Israel said it was tightening the "stranglehold" around Hamas in Gaza City as the military operation on Wednesday aimed at smashing the Palestinian militants gathered pace despite ceasefire calls.
Smoke billowed above the densely-populated coastal territory as fighting raged over a month after the Hamas attacks that killed 1,400 in Israel, sparking the deadliest ever war in Gaza.
According to the Hamas-run health ministry in Gaza, the Israeli military campaign has killed more than 10,300 people, many of them children.
Israel has set an aim of destroying Hamas and said its ground forces were advancing in pursuit of the militants who have a deep network of tunnels and underground bases.
"(Israeli troops) are tightening the stranglehold around the city of Gaza," Defence Minister Yoav Gallant said late Tuesday.
People waving white flags have been fleeing the fighting, while the steadily mounting toll has meant vehicles from donkey-drawn carts to bulldozers have been pressed into transporting the dead.
International concern over the fate of Gaza's civilians, most of whom cannot flee the sealed off territory, has prompted calls for a ceasefire.
Prime Minister Benjamin Netanyahu has said there will be no fuel delivered to Gaza and no ceasefire with Hamas unless the more than 240 hostages seized by Palestinian militants are freed.
G7 foreign ministers said that they supported "humanitarian pauses and corridors" in the Israel-Hamas war but stopped short of calling for a ceasefire.
- 'Death and suffering' -
As fighting intensifies in Gaza, families of Israelis taken hostage by Hamas have been pushing on various fronts for help to bring their loved ones home.
"Every day is like eternity to me and I can't wait any longer," Doris Liber, whose 26-year-old son Guy Iluz was shot and taken hostage at a music festival, told reporters in Washington on Tuesday.
Yonatan Lulu Shamriz, who lived on the Kfar Aza kibbutz on the Gaza border, from which his brother Alon was kidnapped, said: "This is a call for action... this is a wake-up call for all of you here, all of America, all of Europe."
Military analysts warned of weeks of gruelling house-to-house fighting ahead in Gaza, with around 30 Israeli soldiers already killed in the offensive.
The operation is hugely complicated for Israel because of the hostages, including very young children and frail elderly people, who are believed to be held inside a vast tunnel network.
In densely packed Gaza -- where more than 1.5 million people have fled their homes in a desperate search for safety -- the suffering is immense.
The World Health Organization said an average of 160 children are killed every day in Gaza by the war.
"The level of death and suffering is hard to fathom," WHO spokesman Christian Lindmeier said.
Hamas's media office said on Telegram that several cemeteries in Gaza had "no more space for burials", while the UN Office for the Coordination of Humanitarian Affairs (OCHA)said most of the territory's sewage pumping stations were shut.
Israel accuses Hamas of building military tunnels underneath hospitals, schools and mosques -- charges the militant group denies.
OCHA says Israel has ordered all 13 hospitals still operational in northern Gaza to evacuate patients.
- US against re-occupation -
Netanyahu has said Israel will assume "overall security" in Gaza after the war ends, while Ron Dermer, Israel's minister of strategic affairs, said the prime minister was not referring to any future reoccupation of the territory.
Israel withdrew its troops from the territory, which it captured in the 1967 Six-Day War, in 2005.
"After Hamas is removed from power, after we dismantle this infrastructure, Israel is going to have to retain overriding security responsibility indefinitely," Dermer told MSNBC television.
Key ally Washington said it opposed a long-term occupation of Gaza.
Speaking to reporters after G7 foreign ministers held talks in Japan, US Secretary of State Antony Blinken listed what he said were "key elements" in order to create "durable peace and security."
"The United States believes key elements should include: no forcible displacement of Palestinians from Gaza, not now, not after the war; No use of Gaza as a platform for terrorism or other violent attacks; No reoccupation of Gaza after the conflict ends," he said.
In the occupied West Bank on Sunday, Blinken suggested the Palestinian Authority under president Mahmud Abbas should retake control of Gaza.
The PA exercises limited autonomy in only parts of the West Bank, and Abbas said it could only potentially return to power in Gaza if a "comprehensive political solution" is found for the Israeli-Palestinian conflict.
"All over Gaza, helpless people are losing their family members, homes, and their own lives, while world leaders fail to take meaningful action," medical charity Doctors Without Borders (MSF) said.
In its statement, MSF detailed how a staff member was killed on Monday along with his family in Gaza's Shati refugee camp when the area was bombed.
- 'We don't want war' -
Israel has hammered Gaza with more than 12,000 air and artillery strikes and sent in ground forces that have effectively cut it in half.
It has air-dropped leaflets and sent texts ordering civilians in northern Gaza to flee south, but a US official said Saturday at least 350,000 civilians remained in the worst-hit areas.
Clutching one of her toddlers, Amira al-Sakani said she fled Gaza City after coming across the air-dropped Israeli flyers.
"Our life is tragic; we don't want war... we want peace", she added.
The International Committee of the Red Cross, which said one of its humanitarian convoys in Gaza was hit by gunfire on Tuesday, demanded an end to the suffering of civilians.
"Children have been ripped from their families and held hostage. In Gaza, ICRC surgeons treat toddlers whose skin is charred from widespread burns," the organisation's president Mirjana Spoljaric said.
The convoy of five trucks and two Red Cross vehicles was carrying supplies to health facilities, including to Al-Quds hospital, when it was hit, an ICRC statement said, adding that two trucks were damaged and a driver lightly wounded.
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