Left unclaimed is a band through the central part of the city, running along the main east-west route, but US and Iraqi forces are expected to battle to try for that strip later today.
“There’s going to be a movement today in those areas. The heart of the city is what’s in focus now,” said Major Francis Piccoli of the First Marine Expeditionary Force.
this follows warnings that fighting could become even more intense, with several more days of heavy battles expected as forces try to crush insurgents.
The US military said 10 of its soldiers and two Iraqi troops have been killed in the offensive since it bagan on Monday.
Clashes in the city died down late on Tuesday, barely 24 hours after the massive operation was launched to retake the city from rebels.
US marines, backed by crack Iraqi troops, moved down from the north spreading through the city to reach the centre on Tuesday.
On the streets, marines dodged sniper fire and booby-trapped buildings as they battled to secure the city at the heart of Iraq’s insurgency.
Witnesses reported that Jolan, the nerve-centre of the insurrection, had been flattened to rubble and was devoid of civilians or bodies.
The offensive, dubbed Operation Dawn, was launched by Iraqi Prime Minister Iyad Allawi late on Monday aimed at securing the city ahead of key elections in January.
A senior US officer said 20 insurgents had been killed in the Jolan area, about half of them by marine snipers, and that two Egyptians were captured.
While civilian casualties cannot be verified, a man who fled the city told the BBC the streets were littered with bodies.
The officer said three buildings had been found that were wired to explode when the troops entered them, while guns and hand grenades were found in several houses.
US commanders have estimated more than 2,000 fighters, some loyal to Iraq’s most wanted man Abu Musab al-Zarqawi, were in and around the city.
But Lieutenant General Thomas Metz said Zarqawi, blamed for car bombings, kidnappings and beheadings, had probably left Fallujah.
“I personally believe some of the senior leaders probably have fled,” he said. “I hope not, but I have to assume those kind of leaders understand the combat power we can bring and the fact that we will free Fallujah of anti-Iraqi forces.”
Doctors inside the besieged city have painted a grim picture amid a chronic lack of medical equipment, trained staff, water and electricity.
During the offensive the night sky was lit up by tracer fire while the eerie sound of crying babies and hysterical laughter played by US psychological operations forces could be heard.
Meanwhile, hundreds of armed insurgents have taken up positions in the centre of Ramadi as fierce clashes erupted to the northwest of the city in Parawana, and explosions rocked a US base in the city’s east.
And in the country’s north, six Iraqi national guardsmen were killed in a string of roadside bombs near Kirkuk.