Tens of thousands of desperate civilians, some waving white flags, have fled the northern Gaza Strip this week, marching along a road pocked with airstrike craters and lined with ruined buildings, as Israeli officials reported that their troops had pushed into the heart of the densely populated Hamas stronghold of Gaza City.

The United Nations estimated that through Tuesday, 40,000 people had walked out of northern Gaza along the only route south, after Israeli officials told residents of the besieged enclave to evacuate “for their own safety,” and thousands more streamed out on Wednesday.

The evacuations have accelerated in part because of the sharply deteriorating humanitarian situation in northern Gaza, with food and clean water now nearly nonexistent. Israeli forces have also intensified their ground assault in the north, “tightening the noose around Gaza City” and carrying out coordinated attacks deep inside the city, Israel’s defense minister, Yoav Gallant, said in televised remarks on Tuesday night.

As the ground invasion has expanded, the Israeli military said it was offering a daily window of time in which it would guarantee safe passage for civilians.

News photographs as well as a video released by the Israeli military on Wednesday, the fifth day the evacuation corridor was open, showed men, women and children trudging past half-destroyed buildings. Some rode on donkey carts but most were walking, down Salah Al-Din Road, named for the 12th century Muslim sultan and conqueror — known in the West as Saladin — who captured Jerusalem from the Crusaders. They carried small bags, but no suitcases.

The United Nations said that most of those who made the hourslong trip, including children, older people and those with disabilities, traveled on foot with few belongings. It said some arriving in the south reported having to cross Israeli checkpoints, where they said Israeli forces had been making arrests.

Even as many have fled, hundreds of thousands of civilians have stayed in northern Gaza. While some are physically unable to leave, others have said the south seems no safer and that the road leading there is too dangerous.

Foreign ministers of the Group of 7 industrialized democracies, including the United States, called on Wednesday for “humanitarian pauses” in the fighting and corridors in Gaza “to facilitate urgently needed assistance, civilian movement and the release of hostages.” The statement fell short of calling for a cease-fire, which Secretary of State Antony J. Blinken said would risk leaving Hamas in place with the “capacity to repeat Oct. 7 again and again and again.”

Speaking to reporters in Tokyo, where the foreign ministers met, Mr. Blinken said Palestinians should have a government under the Palestinian Authority that unites Gaza and the West Bank. His comments provided the clearest outline yet for what the United States envisions when the war ends.

In recent days, some people who have made or attempted the journey out of northern Gaza have described being fired at from the direction of Israeli tanks, despite the promise of safe passage, while others have reported seeing corpses or body parts around the road. The United Nations is aware of such reports but has not identified who was responsible, according to one U.N. official who asked not to be identified because of the sensitivity of the matter.

On Sunday, Anas Al Kourd, a paramedic at Al Shifa Hospital in Gaza City, drove south with a group that included his cousin, whose legs had been amputated after she was wounded in the war, and four children. Theirs was the only car in sight. Around them, people held their hands up in a gesture of surrender as they walked, carrying little white flags, he said.

As they approached Kuwait Square, a major intersection in Gaza City, they were shot at, forcing them to retreat, he said.

Two or three times they tried to drive forward, only to have to turn around again because of incoming fire. When they finally made it to the square, he said, he could see more than 50 Israeli tanks nearby, where an olive grove used to be. He said he waved a blanket in place of a white flag, while some of his companions waved their red German passports.

Mr. Al Kourd had traveled about a third of the remaining distance to the middle zone of Gaza, bumping over splintered trees and concrete blocks, when he said they were fired on again. He was driving at “mad speed” to get away, he said, when a missile or bomb detonated nearby. No one was struck, but an earlier crater in the road sent their car flying, he said.

“I don’t know how we survived,” he said a day later, having made it to the middle zone of the Gaza Strip. His account could not be independently confirmed.

The Israeli military said in a statement that it has been targeting Hamas throughout Gaza in response to the group’s attack on Oct. 7, when Hamas gunmen killed more than 1,400 people in Israel and seized more than 200 hostages. The statement said that Israel’s attacks on military targets were conducted under international law, including taking “feasible precautions to mitigate civilian casualties.”

The Israeli military denied putting put up checkpoints along the road and maintained that Hamas, not Israel, was making the passage south difficult.

But after a month of heavy Israeli airstrikes, the death toll has been mounting. More than 10,000 people, including more than 4,000 children, have been killed in Gaza, according to the Gaza Health Ministry, which is part of the political arm of Hamas.

The White House has repeatedly said it supports Israel’s right to defend itself and its objective of eliminating Hamas. But it has also steadily increased its calls for humanitarian pauses, increasing pressure on Israel.

On Monday, President Biden called Prime Minister Benjamin Netanyahu of Israel and “discussed the possibility of tactical pauses,” according to a White House statement.

Mr. Netanyahu told ABC News in an interview that aired on Monday that while Israel might consider “tactical little pauses,” there would be no cease-fire without the release of hostages.

Michael Levenson, Nada Rashwan, Hwaida Saad, Motoko Rich, Aaron Boxerman, Victoria Kim and Matthew Mpoke Bigg contributed reporting.