U.S. ground forces invaded Baghdad again today. US officials say they have entered two palace complexes of President Saddam Hussein.

At least two Marines were killed in fierce fighting at key bridges in the eastern part of the city.

Reuters reports explosions continued in central Baghdad, including a residential district. Ambulances raced through the streets.

Information Minister Mohammed Saeed al-Sahaf told reporters Baghdad is safe and claimed US columns had been slaughtered and US forces are committing suicide by the hundreds.

In Saturday’s raid, US commanders claimed allies killed as many as 3,000 Iraqis, but commanders who conducted the raid said around a thousand were killed.

The Pentagon says both raids are shows of force meant to demonstrate to the Iraqi people US forces can enter Baghdad at will.

There was no immediate word on civilian casualties. Aid workers said on Sunday hospitals were already overwhelmed.

The Guardian is reporting thousands of residents are fleeing in cars or on foot, many of them empty-handed or carrying only their children too young to walk in the midday heat.

• May Ying Welsh, independent reporter in Baghdad.