The fiercest fighting of the 13-day war in Gaza has erupted as Israel dramatically widened its ground offensive, sending tanks and troops into urban areas and causing thousands of civilians to flee.

Images of the corpses of women and children were posted on Facebook as hospitals were overwhelmed with the dead, injured and those seeking sanctuary from the onslaught.

People ran from their homes, some barefoot and nearly all empty-handed. Others crowded on to the backs of trucks or rode on the bonnets of cars in a desperate attempt to flee. Some described the bombardment of the Shujai'iya neighbourhood as a massacre.

The Israeli military also suffered casualties, with at least 26 wounded soldiers evacuated. There were unconfirmed reports that Hamas militants had killed a number of Israeli soldiers in a cross-border attack on Sunday morning.

Witnesses reported hearing small arms fire inside Gaza, indicating gun battles on the streets. Heavy shelling continued from the air and sea.

Israel sent more troops into Gaza overnight after demolishing more than a dozen Hamas tunnels and intensifying tank fire on border areas.

Late on Saturday evening, Israeli forces hit eastern areas of Gaza City with the heaviest bombardment yet of the 13-day war. The assault was most intense in the direction of the Shujai'iya neighbourhood, where a constant orange glow of flames lit up the sky.

As the assault continued into Sunday morning, Israel disclosed that four of its soldiers had been killed in the ground offensive.

At one stage, artillery and mortar rounds were hitting the outskirts of the city every five seconds. Later in the night, jets flew low passes over the coast.

As Sunday dawned, a thin pall of smoke hung over the seafront while tank fire echoed through deserted streets.

Large numbers of residents of the areas under attack fled the outskirts for Gaza's city centre, while residents called radio stations pleading for evacuation.

The UN said that more than 60,000 people had sought sanctuary in 49 shelters it was providing in Gaza, and it expected the numbers to rise.

An Israeli air strike on the house of Khalil al-Hayya, a senior Hamas official, killed his son and daughter-in-law and two other children, hospital officials said. Shelling also killed four Palestinians near the southern town of Rafah, officials said.

Lt-Col Peter Lerner of the Israel Defence Forces (IDF) said additional troops had been sent into Gaza on the orders of the government. "Forces have undergone an intensified training and thorough planning period and are prepared and stand ready for the task at hand," he said.

Since the start of Israel-Hamas fighting almost two weeks ago, 348 Palestinians have been killed and 2,700 wounded in Israeli air and artillery strikes, according to Palestinian health officials. A quarter of the deaths had been reported since the start of the ground offensive late on Thursday, they said.

The UN secretary general, Ban Ki-moon, headed to Qatar on Sunday as part of renewed ceasefire efforts. He was due to meet the Palestinian president, Mahmoud Abbas, in Doha.

Abbas was also expected to meet Khaled Meshaal, the Hamas leader based in the Qatari capital.

According to the Egyptian newspaper Aharam, the US secretary of state, John Kerry, was travelling to Cairo to help in the mediation effort.

Hamas last week rejected an Egyptian call for both sides to halt hostilities, insisting on advance guarantees that Israel and Egypt would significantly ease their border blockade of Gaza. Qatar has presented a ceasefire proposal incorporating Hamas's demands, but Egypt said on Saturday that it had no plans to revise its proposal.

Israel is opposed to Qatar's involvement, and insists that Egypt must be a party to any deal. Doha hosts a large number of exiled Islamists from across the Middle East, including Meshaal.

The French foreign minister, Laurent Fabius, who flew to Israel after meetings in Egypt and Jordan, said on Saturday that efforts to secure a ceasefire had failed. "Sadly I can say that the call for a ceasefire has not been heard, and on the contrary, there's a risk of more civilian casualties that worries us," he said following talks with the Israeli prime minister, Binyamin Netanyahu.

According to the Israeli military, its soldiers have uncovered 34 shafts leading into about a dozen underground tunnels, some as deep as 30 metres. Israel has said demolishing tunnels is the principal goal of its ground operation, and it released footage of them being demolished by excavators and air strikes.

The IDF reported three cross-border incidents on Saturday. The most serious involved 12 Palestinian militants disguised in Israeli uniforms who emerged from a tunnel in Israel to fire an anti-tank missile at Israeli troops, killing two and injuring several others.

They were "aiming to carry out a lethal attack" on a nearby Israeli community, the IDF said. The dead soldiers were named as Bar Rahav, 21, and Benayahu Rubel, 20.

At least one Palestinian was killed in the clash. Hamas said its fighters had taken some of the soldiers' weapons back to their hideouts.

In other confrontations, Palestinian gunmen emerged from tunnels and exchanged gunfire with Israeli soldiers. Two of the militants were killed, and another died when the explosive vest he was wearing went off, the military said.

In one incident, Hamas fighters carried tranquilisers and handcuffs, indicating they "intended to abduct Israelis", according to the military.

As the offensive intensified, electricity and water supplies in Gaza were increasingly disrupted.

The Gaza City municipality said a water main had been damaged, leaving parts of the city without supplies. Gaza has suffered from rolling blackouts for years, but periods without electricity have now increased to up to 20 hours a day.