Israel's prime minister has warned of a "prolonged" military campaign in Gaza, as it saw one of its heaviest nights of shelling since the conflict began.
Benjamin Netanyahu said Israel would continue to act until it had achieved its aim of destroying militant group Hamas's network of underground tunnels.
Israel made 60 air strikes on targets in Gaza overnight including TV stations and the house of a key Hamas leader.
It said militants had launched three rockets at Israel.
At least 13 Palestinians are reported to have been killed in the overnight attacks, including six victims in one house in the Bureji refugee camp, while Israel's army lost 10 soldiers in the past 24 hours.
Officials say more than 1,100 Palestinians, most of them civilians, have been killed in the fighting since 8 July. Israel says 53 of its soldiers and three civilians - two Israelis and a Thai worker, have been killed.
Early on Tuesday, Israeli aircraft fired at the unoccupied house of former Hamas Prime Minister Ismail Haniyeh, while Hamas TV and radio stations were also hit.
Three rockets were fired from Gaza, with one rocket successfully intercepted and two hitting open spaces in central Israel, Israel's military said.
At least 10 people - eight of them children - were killed in blasts in Gaza City on Monday afternoon, Palestinian health officials said. It is unclear if they were killed by an Israeli attack or a misfiring militant rocket.
Five Israeli soldiers were killed on Monday when militants infiltrated the border, while a mortar bomb killed four earlier and a tenth died in a clash in southern Gaza, the Israel Defence Forces (IDF) said.
Mr Netanyahu described Monday as a "painful day". "We will continue to act aggressively and responsibly until the mission is completed to protect our citizens, soldiers and children," he said.
On Monday, UN Secretary General Ban Ki-moon criticised both sides for firing into civilian areas, and called for an immediate, unconditional humanitarian ceasefire in Gaza during the Muslim Eid al-Fitr holiday.
His spokesman later added that Mr Ban was concerned at reports that leaflets had been dropped by the IDF warning residents in the northern Gaza Strip to evacuate to Gaza City.
"If true, this would have a further devastating humanitarian impact on the beleaguered civilians of those areas of the Gaza strip, who have already undergone immense suffering in recent days," his spokesman said.
Rocket fire and air strikes between the two sides increased after the abduction and killing of three Israeli teenagers in June, which Israel blamed on Hamas and which led to a crackdown on the group in the West Bank. Hamas denied being behind the killings.
Tensions rose further after the suspected revenge killing of a Palestinian teenager in Jerusalem on 2 July. Six Jewish suspects were arrested over the youth's abduction and murder.
On 8 July, Israel launched Operation Protective Edge, an offensive against Hamas in Gaza, after a surge in rocket fire.
On 18 July, it extended operations with a ground offensive, saying it was necessary to destroy tunnels dug by militants to infiltrate Israel.
Since then, there have been hundreds of air strikes and hundreds of rockets have been fired.
The Gaza Strip, sandwiched between Israel and Egypt, has been a recurring flashpoint in the Israel-Palestinian conflict for years.
Israel occupied Gaza in the 1967 Middle East war and only pulled its troops and settlers out in 2005. Israel considered this the end of the occupation, but it still exercises control over most of Gaza's borders, waters and airspace. Egypt controls Gaza's southern border.