As the US military seized what it said was more than two-thirds of the city by early Thursday, Iraqi troops fighting alongside them said they found houses where foreign captives had been held and, in some cases, killed.
Machine gun, mortar and rocket fire shook the city as US warplanes made several bombing runs over the Julan district in the space of 15 minutes.
With US marines backed by Iraqi troops moving from building to building, a US officer said they expected to take command of the city before the weekend.
"If everything goes as planned we will take full control of the
city in the next 48 hours," he said on condition of anonymity, adding that it would take "at least 10 days to clear the city".
But Abu Shams al-Fallujy, a member of the National Islamic Resistance in Falluja, told Aljazeera that US forces had entered the city's centre rapidly because they were surrounded in the Julan neighbourhood and were being targeted by snipers.
"The US troops say they have controlled the town. In fact, they are only in control of the town centre where there are no resistance fighters. The town's centre is resided by the civilian population who had escaped the outskirts of the town to avoid a crossfire as a result of the intense US bombing and resistance.
US troops have encountered
fierce resistance in the city
"With respect to Julan, al-Askary, Nizal, Jibail and the industrial quarter, they are still under control of the resistance which vows not to allow the US forces to control the town unless it is turned into dust," al-Fallujy said.
"The situation in the town is very critical. The US forces began a retreat under intense resistance fire. They are conducting a ferocious aerial bombing and artillery barrage. They have not accomplished any advance towards the edges of the town," al-Fallujy added.
Signs of the killings of captives were found by Iraqi troops in the northern districts, according to an Iraqi general.
"We have found hostage slaughterhouses in Falluja that were
used by these people and the black clothing that they used to wear to identify themselves, hundreds of CDs and whole records with names," Major-General Abdul Qader Mohan said.
Acting on the orders of US-appointed interim Iraqi Prime Minister Iyad Allawi after weeks of heavy US aerial and artillery bombardment, about 10,000 US and 2000 Iraqi soldiers stormed Falluja late on Monday.
Allawi's kin captured
Allawi has vowed to crush the rebellion ahead of elections
planned for January.
Allawi's cousin, his wife and
son have been captured
But the tough-talking premier's family got caught up in the battle when a gang in three cars snatched his cousin, Ghazi Allawi, his wife and their son in Baghdad overnight, spokesman for the interim government Thaer al-Nagib said.
They were captured in front of their home in Yarmuk district in Baghdad.
A previously unknown group, Ansar al-Jihad, threatened in an internet statement to kill all three within 48 hours unless Allawi halted the Falluja attack and released all Iraqi prisoners.
Scores of fighters have been killed and at least 11 US and two Iraqi soldiers have died in the offensive, the US military said.
But al-Fallujy told Aljazeera: "The toll among the resistance is very scant. Most of the dead are civilians in the town centre in al-Dhubbat and al-Wuhda neighbourhoods.
"There are so many dead and wounded civilians who need medical assistance. But the physicians and the ambulances cannot reach these people because of the US sniper fire," he added.
No medical help
The Red Cross and Red Crescent expressed concern over the fate of the wounded, as the government said some of the tens of thousands who fled were ill and living in cramped conditions.
Up to half the population of 300,000 may have stayed
In an effort to persuade fighters in the Sunni Muslim city to lay down their arms, Allawi offered an amnesty to those who had "committed no major crimes".
The assault on the city - where residents say wounded children are dying from lack of medical help, food shops are closed and power has been cut - has angered Muslim clerics.
And in a move that could potentially undermine the planned 27 January elections, the Association of Muslim Scholars (AMS), urged a boycott.
"The clerics call on the ... people of Iraq to boycott the coming elections that they want to hold on the remains of the dead and the blood of the wounded from Iraqi cities like Falluja and others," Harith al-Dhari, its top official, said.
The AMS also reported that one of its members, Shaikh Abdul Wahab al-Janabi, was killed in the Falluja attack.
Iraqi journalist Fadil al-Badrani said many civilians had died in the bombing of the city and people had resorted to burying their dead in gardens. Many houses have also been destroyed.