Israel said it killed 16 members of the Hamas armed wing in Gaza, including the brigade commander of Gaza, in a barrage on Wednesday.
Palestinian militants also rained rockets into Israel, killing six people.
The clashes continued as Washington said it would send an envoy to try to calm their most intense hostilities in years.
At least 56 people have been killed in Gaza so far since violence escalated on Monday, according to the enclave’s health ministry.
Israeli Prime Minister Benjamin Netanyahu, who confirmed the death of the brigade commander said:
“This is just the beginning. We’ll hit them like they’ve never dreamed possible,” he said.
Soon after the announcement, a new rocket barrage was fired at the Israeli city of Ashdod and Israeli media said the military was preparing for new salvoes on the Tel Aviv area.
Hamas confirmed the death of the commander and of “other leaders and holy warriors” in a statement.
Its leader Ismail Haniyeh added: “The confrontation with the enemy is open-ended.”
Among those killed were two Hamas intelligence officers.
A Palestinian source said truce efforts by Egypt, Qatar and the United Nations had made no progress to end violence that flared this week after tension during the Muslim fasting month of Ramadan and clashes in East Jerusalem.
Describing the scenes of destruction as “harrowing”, U.S. Secretary of State Antony Blinken said a senior aide, Hady Amr, would be sent to urge Israelis and Palestinians to seek calm.
Pentagon chief Lloyd Austin reaffirmed “ironclad support for Israel’s legitimate right to defend itself”. Israeli Defence Minister Benny Gantz said Israel would continue to strike Hamas to restore long-term calm, according to a statement on their call.
The fighting is the heaviest between Israel and Hamas since a 2014 war in the Hamas-ruled enclave, and concern is growing that the situation could spiral out of control.
In Gaza, a multi-storey residential building and a tower housing Hamas and other media outlets collapsed after Israel warned occupants in advance to evacuate, and another structure was heavily damaged in the air strikes.
Other air strikes hit what Israel’s military said were rocket launch sites, Hamas offices and homes of Hamas leaders.
“Israel has gone crazy,” said a man on a Gaza street, where people ran out of their homes as explosions rocked buildings.
Twenty-four people were killed in Israeli air strikes on Gaza on Wednesday, Gaza’s health ministry said. Witnesses and health officials in Gaza said one Israeli air strike killed three people, including a woman, in a car.
Many Israelis also spent a sleepless night as waves of rockets hit Israel’s heartland.
“The children have escaped the coronavirus, and now a new trauma,” an Israeli woman in the coastal city of Ashkelon told Channel 11 TV.