US commanders say they are trying to trap insurgents in Falluja against the River Euphrates in the southern part of the Iraqi city.
Marines say they are gaining control of the city street by street
The American-led assault on the city is in its fifth day.
A BBC correspondent in the city centre says US marines are still under sniper attack at their main base.
Militant leader Abu Musab al-Zarqawi has allegedly urged rebels to resist US forces in Falluja in a recording posted on an Islamist website.
US troops say they have trapped fighters loyal to al-Zarqawi in the south of Falluja but the recording assures insurgents that victory is near.
The BBC's correspondent in Falluja says the insurgents are increasingly confined to a narrow corridor south of the main road through the city.
Lt Col Gareth Brandl of the US marines told our correspondent that the gradual driving back of the insurgents was working.
"We are here to reduce the enemy sanctuary, we're doing that. We are crushing his back, one vertebrae at a time."
Embedded with US forces and subject to reporting restrictions, the BBC's Paul Wood says there is no sign of any civilians and the plight of those trapped by the fighting can only be guessed at.
US forces estimate they have killed about 600 insurgents since the assault began on Monday. They say 18 Americans have been killed and almost 180 wounded.
In other developments in Iraq:
- A US Blackhawk helicopter is shot down north of Baghdad, wounding three crew members
- Insurgents shoot dead an American soldier after attacking a US patrol in southern Baghdad; clashes continue in other parts of the capital
- The US launches air strikes against targets in Mosul after fierce battles with insurgents
- In Falluja, US forces say they have found alive the Syrian driver who was kidnapped with two French journalists in Iraq almost three months ago. There is no information on the Frenchmen.
Master Sergeant Roy Meek of the US marines said the insurgents were cornered in the south of the city.
"They can't go north because that's where we are. They can't go west because of the Euphrates river and they can't go east because we have a huge presence there."
US Corporal Will Porter said he was charged with clearing insurgents from one house at a time.
"It's extremely dangerous right now because the insurgents have
nowhere to go, they are just sitting in houses waiting for us to
come in," he said.
US officials nearly doubled their tally of wounded on Friday, and bed capacity is reportedly being expanded at the main US military hospital in Europe - at Landstuhl in Germany - to cope with an influx of injured marines.
Some 10,000 US forces and 2,000 Iraqi troops are involved in the battle for Falluja.
The Iraqi Red Crescent said the humanitarian situation in Falluja was a "disaster".
"Anyone who gets injured is likely to die because there's no
medicine and they can't get to doctors," said Abdul-Hameed
Salim, a volunteer.
Residents trapped in the battered city said they could smell the stench of decomposing bodies.
Rasoul Ibrahim, who fled Falluja on foot with his wife and three children on Thursday, said conditions there were desperate.
"There's no water. People are drinking dirty water. Children are dying. People are eating flour because there's no proper food," he said.