But the operation was disavowed by Prime Minister Nouri al-Maliki.
Al-Maliki, who relies on political support from the cleric Muqtada al-Sadr, said the strike against a figure in al-Sadr's Mahdi militia in Sadr City "will not be repeated."
But in Sadr City, protesters shouted "Maliki, you're a coward, you're the Americans' traitor," reports CBS News chief foreign correspondent Lara Logan.
Military incursions into Sadr City are never welcome. The sprawling Shiite slum in the northeastern corner of the capital is home to 3 million people, half the population of Baghdad.
The problem for al-Maliki is that Sadr City's streets belong to al-Sadr and his militia. In Sadr City, he's the people's hero.
Reining in al-Sadr's Mahdi army is one of the thorniest problems facing al-Maliki because his fragile, Shiite-dominated government derives much of its power from al-Sadr.
The Mehdi army helped put al-Maliki in power, and just last week he sought al-Sadr's help again — to rein in the violence.
Al-Maliki also slammed the top U.S. military and diplomatic representatives in Iraq for saying his government needed to set a timetable to curb violence in the country. At a news conference Tuesday, U.S. Ambassador Zalmay Khalilzad said al-Maliki had agreed.
"I affirm that this government represents the will of the people, and no one has the right to impose a timetable on it," al-Maliki said at a news conference.
The prime minister dismissed U.S. talk of timelines as driven by the upcoming midterm elections in the United States. "I am positive that this is not the official policy of the American government, but rather a result of the ongoing election campaign. And that does not concern us much," he said.
In Washington, President Bush sought to delineate a middle ground in terms of pressing the Iraqis to accept more responsibility for their own fate.
"We are making it clear that America's patience is not unlimited," he said. "We will not put more pressure on the Iraqi government than it can bear."
Tank cannons boomed out over the city five times in rapid succession Wednesday, and U.S. F-16 jet fighters screamed low overhead as the conflict in Sadr City continued into the day.
Four people were killed and 18 wounded in overnight fighting in the overwhelmingly Shiite eastern district, said Col. Khazim Abbas, a local police commander, and Qassim al-Suwaidi, director of the area's Imam Ali Hospital.
Iraqi army special forces, backed by U.S. advisers, carried out a raid to capture a "top illegal armed group commander directing widespread death squad activity throughout eastern Baghdad," the military said.
Al-Maliki, who is commander in chief of Iraq's army, heatedly denied he knew anything about the raid.
"We will ask for clarification about what has happened in Sadr City. We will review this issue with the multinational forces so that it will not be repeated," he said. "The Iraqi government should be aware and part of any military operation. Coordination is needed between Iraqi government and multinational forces."
Mr. Bush acknowledged that al-Maliki may not have been consulted beforehand.
"There's a lot of operations taking place, which means sometimes communications are not as good as they should be. And we'll continue to work very closely with the government to make sure communications are solid," he said.