Sadr City has seen the heaviest fighting for weeks
Four US soldiers have been killed in two separate mortar or rocket attacks in Baghdad, the latest casualties of fierce clashes with Shia militia.
At least 38 Shia militia fighters have also been killed in the past two days of fighting, the US military has said.
Twenty-two militants died in a single incident on Sunday, when US tanks opened fire to repel an attack.
A US military spokesman said the operation in Sadr City was targeting criminals and militants firing rockets.
Three US soldiers were killed by indirect fire - the US military term for mortar or rocket fire - in eastern Baghdad on Monday. Another soldier was killed by a similar attack in northern Baghdad.
Their deaths, according to Iraq Coalition Casualty Count, bring the total number of US soldiers killed to 4,055 since the US-led invasion in 2003.
"When we have losses like this they hit us hard, but they bring us together, they strengthen our resolve and we will win," Major Mark Cheadle of the Multi-National Force in Baghdad told the BBC.
"We are taking it to them every single day. They are using dastardly methods of hiding behind civilians... But when they face us head to head, as the incident on Sunday showed, they lose every time."
Sadr City is a Baghdad stronghold of radical Shia cleric Moqtada Sadr.
Militants had taken advantage of a sandstorm on Sunday to shell the heavily protected Green Zone, officials said.
Local residents stayed indoors and aerial military operations by US forces were initially made difficult by the sandstorm, the BBC's Clive Myrie reports from Baghdad.
The Green Zone, the vast complex housing government offices and foreign embassies, was still being hit by rocket or mortar attacks on Monday.
US and Iraqi officials have blamed such attacks on rogue elements from the Mehdi Army, a militia loyal to Moqtada Sadr.
The cleric called for a ceasefire on Friday, but on Sunday he rejected the government's conditions for ending a major military campaign against Shia militias.
Major Cheadle told the BBC the upsurge in violence followed an operation to end rocket attacks on the capital's International Zone.
"We've established a combined civil-military operations centre with representatives from the government of Iraq," he said.
"We've been there since actually about October but we just started to clear the areas of criminals and those who are attacking with rockets inside the IZ [International Zone - the military's term for the Green Zone] and hurting innocent civilians in and around the IZ."
Iraqi Prime Minister Nouri Maliki has said he wants to disarm all militants operating outside the control of the central government.
More than 400 people have been killed in Sadr City in recent weeks.