Israeli forces are advancing on Gaza City, Hamas’s main stronghold, from three directions

Israeli forces on Monday advanced from three directions toward the battered and densely populated Hamas stronghold of Gaza City, slowly pushing deeper into the Gaza Strip while battering the territory with airstrikes and renewing evacuation warnings to people and hospitals.

On the fourth day of Israel’s invasion, formations of its troops and armored vehicles approached the city and neighboring population centers from the north, east and south, as shown by eyewitness photos and videos verified by The New York Times, as well as satellite imagery.

The deepest penetration, about three miles, paralleled the Mediterranean coast toward the Shati refugee camp on the northern flank of the city. South of the city, Israeli forces reached Salah Al-Din Road, one of the main north-south arteries used by people trying to heed the Israeli call to evacuate northern Gaza. One widely circulated video showed an armored vehicle there firing on a car, which Palestinians said could discourage people from attempting the trip.

Prime Minister Benjamin Netanyahu and other Israeli leaders have vowed to wipe out Hamas, the armed group that controls Gaza, after its Oct. 7 attack on Israel. According to the Israeli government, Hamas-led fighters killed more than 1,400 people, mostly civilians, and seized more than 230 hostages, including women and young children.

“Israel is fighting the enemies of civilization itself,” Mr. Netanyahu said at a news conference on Monday with foreign news outlets. He rejected the idea of a cease-fire, backed by humanitarian groups and the United Nations General Assembly, among others, adding, “calls for a cease-fire are calls for Israel to surrender to Hamas, to surrender to terrorism.”

Israel has responded to the Oct. 7 assault with a ferocity far beyond its previous conflicts with Hamas, first with an intensive air and artillery bombardment that is still underway, with hundreds of strikes daily. The campaign has killed more than 8,000 Gazans, according to the Hamas-controlled Gaza Health Ministry, displaced more than a million and physically devastated large swaths of the Palestinian territory.

Some people still in northern Gaza said Israel’s instruction to evacuate meant nothing to them because they had no place to go and no safe way to travel, or simply because people were dying in the south, too.

“If they invade on the ground, we will die at home,” Bilal Assabti, a 30-year-old unemployed father of two young children, said by phone on Monday evening. “We do not have anyone in the south. Where should we go?”

Israel faces growing international criticism for its conduct of the war, including the evacuation directive and strikes on civilian targets that Israel says are used to conceal Hamas troops, command posts, tunnels and supply depots. Hospitals, schools and shelters have also been damaged, and UNRWA, the main United Nations agency that provides humanitarian assistance to Palestinian refugees, said that 63 of its employees there have been killed since Oct. 7.

“We’re going out of our way to prevent civilian casualties,” Mr. Netanyahu said in his news conference, held in English, which was framed as an appeal to the world to support Israel in fighting Hamas. He called on other countries to recognize “a moral distinction between the deliberate murder of the innocent and the unintentional casualties that accompany every legitimate war, even the most just war.”

He drew a parallel to a British bombing raid on Copenhagen in World War II that destroyed its intended target, the Gestapo headquarters in occupied Denmark — and also struck a school, killing 86 people there.

President Biden has not publicly found fault with Israel, but has urged restraint. On Sunday he reiterated support for Israel’s right to protect itself, according to the White House, while underscoring “the need to do so in a manner consistent with international humanitarian law that prioritizes the protection of civilians.”

Israel continued to warn hospitals in northern Gaza to evacuate, the World Health Organization said overnight. But the W.H.O. said it was impossible to move without risking patients’ lives, and health officials say there is nowhere for them to go, with some hospitals shut down and the remaining ones already overcrowded and dangerously short of essential supplies.

The chief spokesman for Israel’s military, Rear Adm. Daniel Hagari, declined to say how many Israeli troops were inside Gaza, or where, but confirmed that a combined force of infantry and armored units, operating with air support, was engaging in “expanded ground operations,” but was moving gradually. He said Hamas gunmen typically gather at “staging sites” before trying to attack Israeli soldiers, “after which we strike them from the air.”

Overnight, he added, “dozens of terrorists were eliminated” after they barricaded themselves inside buildings and attempted to attack the soldiers who were moving in their direction. It was not possible to verify Israel’s account of the fighting.

Hamas’s armed wing released a video showing three women who were kidnapped on Oct. 7; one of them sharply criticizes Mr. Netanyahu, saying the hostages are being held in “unbearable conditions” and demanding that he exchange them for Palestinian prisoners held by Israel.

“When I saw Daniel on television, my heart almost stopped,” said Ramos Aloni, the father of Daniel Aloni, one of the hostages in the Hamas video. “My wife and I were shocked, but we also felt relief — she was alive and we were seeing her.”

Mr. Netanyahu has been under pressure from the families of some hostages held in Gaza, who have accused his government of prioritizing the military campaign over the effort to bring back their loved ones. Some have even expressed willingness to consider a deal that would exchange Israel’s Palestinian prisoners for the hostages.

Mr. Netanyahu’s office called the Hamas video “cruel psychological propaganda,” and in his news conference, the prime minister said the invasion of Gaza “actually creates the possibility of getting our hostages out.”

The Israeli military also said it had rescued a woman, a 19-year-old soldier taken hostage in the Hamas incursion and held in Gaza — the first known rescue operation since the mass abduction on Oct. 7 — as its forces continued their ground invasion there.

Conditions are dire for civilians in Gaza, under an Israeli and Egyptian blockade. Forty-seven trucks carrying food, water, medical supplies and other humanitarian aid entered Gaza through the Rafah crossing with Egypt on Sunday, according to a Palestinian official at the crossing. That was the largest one-day total in the nine days since the shipments began, but less than half of what the United Nations says is needed.

Israeli and Western officials say Hamas has stockpiled ample supplies but is hoarding them for itself rather than putting them to humanitarian use — a charge the group denies. Fuel is increasingly scarce in Gaza, leaving most people without electricity and hobbling hospitals and shelters, but Israel has refused to allow fuel into the territory, arguing that it would be diverted to Hamas.

Clashes continued on Israel’s other flanks, as much of the world watches warily for signs of a broadening war. In the Israeli-occupied West Bank, at least four people were killed overnight during an Israeli military raid in Jenin, the Palestinian Authority health ministry said, where Israel said its troops had engaged in a gun battle with Palestinian militants. At least 115 Palestinians in the West Bank have been killed since Oct. 7 by Israeli security forces or settlers.

The Israeli military said it had also traded fire overnight with enemies across its northern border, in Lebanon and Syria.