Israeli troops had sent ground troops into the Shajaiye neighborhood late last month and the fighting has since expanded to other parts of the city. The military said later on Wednesday that it had concluded its operation in Shajaiye, but it gave no indication that fighting would end elsewhere.

Israeli troops have re-entered Gaza City in recent days, in the latest instance of Israeli forces returning to fight in places they had secured earlier and then withdrew. The Israeli military has repeatedly returned to areas across the Gaza Strip in an attempt to suppress Hamas fighters, who have fought a dogged guerrilla war. Analysts have said Israel’s unwillingness to install an alternative administration in Gaza has created a power vacuum, allowing Hamas to regroup.

In January, the Israeli military dialed back the intensity of its military campaign in Gaza City and the rest of the north. Since then, Israeli forces have carried out a series of targeted raids in the area, and in March its troops raided Al-Shifa hospital for a second time, killing nearly 200 people it called “terrorists” and leaving devastation behind after extended gun battles with Palestinian militants.

It is not clear how many Hamas fighters remain in Gaza City. After sending ground forces into Shajaiye, Israeli troops moved into other parts of the city: Tel al-Hawa, where Israeli forces stormed a United Nations compound that the military said had taken over by militants, as well as the neighborhoods of Al-Daraj and Tuffah.

In statements on social media, Hamas has said over the past few days that its forces were fighting Israeli troops in Shajaiye and Tel al-Hawa. In Shajaiye alone, Israel claims its troops have eliminated “more than 150 terrorists” over the past week and have destroyed six underground tunnels.

Hamas has used urban areas in Gaza to conceal its operations, running tunnels under neighborhoods and holding hostages in city centers. The group’s members, who are from Gaza, have long lived among the civilian population.

Ghazi Hamad, a senior Hamas official, has said that the group tries to keep Palestinian civilians out of harm’s way, but Israeli operations in Gaza have left little room to maneuver.

Israel first ordered hundreds of thousands of Gazans in the northern part of the enclave to move south in mid-October, just days after the Hamas-led attack that killed 1,200 in Israel and saw 250 taken hostage. Hundreds of thousands remained, however, and others joined them after a weeklong truce in November allowed some to return to their homes in the north.

In May, at least 200,000 people were still in northern Gaza, according to the United Nations. But the new wave of Israeli military operations has forced tens of thousands from their homes, leaving the current tally unclear.

Many have been already been displaced multiple times, seeking shelter in schools and relatives’ homes, only to be forced to flee the fighting yet again.

“People continue to flee and be on the run in search for safety that they never find,” said Juliette Touma, a spokeswoman for UNRWA, the United Nations agency for Palestinians, said on Wednesday. “Gaza has become an exodus on repeat.”

In the video, shot at the Al Awda School on the outskirts of Khan Younis, the ball is in midair when a large explosion is heard and the camera shakes. A man yells, “Run away, run away, Al Awda has been targeted!”

The person shooting the video runs to the entrance of the school, and the camera pans across a scene of devastation. Shredded bodies are on the ground amid debris, and there is a cacophony of screams. “Oh God,” someone yells.

The Israeli military said that the strike targeted a Hamas member who took part in the Oct. 7 attacks on Israel that ignited the war. It did not release details on the identity of the Hamas member or whether the person had been killed. The military said it was “looking into reports that civilians were harmed.”

Iyad Qadeh, who was sitting outside his home near the entrance of the school at the time of the strike, said it had been calm and there were not even Israeli surveillance drones in the sky, as there often are above Gaza, creating a nearly nonstop buzzing.

Suddenly, a warplane flew overhead and fired a missile toward a group of young men sitting in an internet café, he said.

“After that, it was screams and body parts everywhere,” he said. “Everyone started running searching for their children or family members.”

The strike was the fourth in four days that hit or damaged a school building in Gaza, Philippe Lazzarini, commissioner-general of UNRWA, the U.N. agency that helps Palestinians, wrote on social media Wednesday.

Since Israel began its punishing military offensive in Gaza more than nine months ago, two-thirds of U.N.-run school facilities in the territory have been hit, Mr. Lazzarini wrote. “Schools have gone from safe places of education & hope for children to overcrowded shelters and often ending up a place of death & misery,” he said.

Those shelters have become critical for Palestinians in Gaza since Israeli bombardment and ground fighting have forced much of the territory’s 2.2 million residents to flee their homes. The Israeli military has claimed that militants are using such shelters and other civilian buildings to hide themselves and their activities.

Ghazi Hamad, a senior Hamas official, has said that the group tries to keep Palestinian civilians out of harm’s way.

Most of those injured or killed in the strike on Tuesday were taken to Nasser Hospital in Khan Younis in ambulances, private vehicles and donkey carts. Dr. Mohammed Saqer, director general of nursing at Nasser, said in a phone interview on Wednesday that 56 people had been wounded, most of them children and women. “And unfortunately nearly 10 cases of amputation among them,” he said, “hands and feet completely blown off.”

The state of the bodies brought to Nasser made it difficult to determine the number and identities of the dead, he added.

The influx of traumatic injuries came at a time when the few still-functioning hospitals in the Gaza Strip are struggling to keep running amid Israeli strikes and raids and a lack of medicine, medical equipment and reliable power. “Many of our medical staff have been detained, many have been killed and many have had to leave Gaza,” Dr. Saqer said.

There is a shortage, too, of hospital beds, and most of the airstrike victims were treated on the floors of wards or in the hallways, he said.

A video shot by the Reuters news agency at the site of the strike showed the fragment of a weapon. Two weapons experts — Trevor Ball, a former U.S. Army explosive ordnance disposal technician, and Patrick Senft, a weapons expert at the consulting firm Armament Research Services — identified the fragment as a part of a small-diameter bomb, also known as a GBU-39.

The precision-guided bomb, which is U.S.-made, weighs about 250 pounds, and is increasingly the weapon of choice for the Israeli military. Two GBU-39s were used in a deadly strike on a tent camp in Rafah on May 26.

In Gaza, such bombs “are often used to target specific floors in buildings, penetrating through the roof before detonating,” Mr. Ball said.

Although smaller in explosive power than the 2,000-pound bombs that have been used elsewhere in Gaza, the bombs “can still cause significant injury and death, especially when used in areas where there is little to no protection for people from blast and fragmentation effects, such as a street, or area with just tents,” he said.

Malachy Browne, Sanjana Varghese and Ameera Harouda contributed reporting.

“Hamas is negotiating on its own behalf and on behalf of the entire resistance axis, and what Hamas accepts, we accept,” Mr. Nasrallah said in a televised speech.

Israeli forces and Hezbollah fighters in Lebanon, who are backed by Iran, have traded fire since the war in Gaza began after Hamas led an attack on Israel on Oct. 7. About 150,000 people in northern Israel and southern Lebanon have fled their homes because of the fighting, and world leaders are worried that continued hostilities could quickly spiral into a full-fledged war that further destabilizes the already fraught region.

Mr. Nasrallah, in his speech, referred to these concerns, noting that “many delegations” from the international community have visited Lebanon to discuss defusing tensions: “We repeated the same words: If you want to stop the northern front, stop the fire in Gaza.”

But the cease-fire talks have been halting, and the fighting between Israel and Hezbollah — cross-border attacks with drones, rockets and airstrikes — has escalated in recent weeks, heightening fears that a wider war may be imminent.

On Tuesday, two Israelis — Noa and Nir Baranes, a married couple from Kibbutz Ortal in the northern Golan Heights — were killed in a Hezbollah strike on their car that the Lebanese armed group said had targeted an Israeli military base in response to an earlier Israeli “assassination” of a Hezbollah figure in Syria. The civilian deaths put additional pressure on the Israeli government to address tensions with Hezbollah, even as the Israeli military keeps returning to parts of Gaza it had previously considered pacified to suppress a resurgence of Hamas fighters.

The Israeli military said on Wednesday that its air force struck a “military site” in southern Lebanon after soldiers identified several Hezbollah operatives entering it, and that it also had targeted other Hezbollah sites nearby. Israeli government officials and military leaders have in recent weeks toured northern Israel and met with troops stationed there who were preparing for a potential escalation.

Israel has invaded Lebanon three times in the last 50 years, most recently in 2006, when the sides fought a monthlong war that killed more than 1,000 people in Lebanon, mostly civilians, and more than 150 in Israel, mostly soldiers.

Yoav Gallant, Israel’s defense minister, has said that a diplomatic solution to the conflict with Hezbollah is preferable. But he has also emphasized, including in talks with his U.S. counterpart, Lloyd J. Austin III, in Washington late last month, that Israel is “determined to establish security” in the north and change “the reality on the ground.”

Israeli security experts are concerned that a war against Hezbollah will be more intense — and more likely to draw in other players — than the fight against Hamas.

A new report from Tel Aviv University’s Institute for National Security Studies warned that “Hezbollah has the military capabilities to conduct an exceedingly protracted war, probably lasting many months, and cause severe damage to Israel” and that it could turn into “a multi-front war against Iran and its other proxies.”

The report, led by the retired Israeli Brig. Gen. Meir Elran, a senior researcher at the institute, said “there is a growing sense of futility regarding the future of the northern border.” It also noted that “public discourse has been heavily focused on the possibility of a comprehensive war with Hezbollah.”

Comprehensive war is precisely what France, the United States and others have been hoping to prevent with diplomatic visits to the region. Whether efforts at de-escalating the conflict between Israel and Hezbollah can succeed may well depend on how the cease-fire talks in Qatar progress.

On Wednesday, the C.I.A. director, Mossad chief and Egyptian intelligence head met in Qatar with the Qatari prime minister in an effort to find a peace formula both Israel and Hamas can agree on. (Egypt and Qatar have been relaying proposals to Hamas leaders.)

So far, however, Israel and Hamas disagree over a fundamental question: whether the cease-fire will be permanent or just a temporary pause to allow for an exchange of hostages taken from Israel for Palestinian prisoners.

The talks have been “progressing positively,” said Majed al-Ansari, the spokesman for Qatar’s Foreign Ministry, before the latest gathering. But, he added, “We are by no means out of the woods.”