An Iraqi journalist told Aljazeera that US forces on Wednesday resumed attacks on the city, targeting Julan in the north-west to al-Jughaivi in the north-east.
Journalist Fadil al-Badrani said there are an estimated 120 mosques in the city.
"Almost half of the city's mosques have been destroyed after being targeted by US air and tank strikes," he added.
Fierce clashes also erupted as US forces moved deeper into the city in the early hours, he said.
Machine gun, mortar and rocket fire shook the city as warplanes made several bombing runs over Julan district in the space of 15 minutes.
Smoke was rising from houses just beyond Falluja's captured railway station, where marines and Iraqi forces have a base.
Marines said their opponents showed no signs of giving up, even though US forces penetrated to the centre of the city, west of Baghdad, after an offensive launched on Monday night.
A tank platoon that moved along Falluja's main street saw fighters who had just come under mortar fire climb on to rooftops and fire rocket-propelled grenades (RPGs) and machine guns.
"There are lots of them. We took heavy fire," Gunnery Sergeant Ishmail Castillo said. "They opened up on my tank. They don't look like they are going to cave in."
Castillo said his tank had killed six fighters and that two marines were wounded in fighting. "One of the marines was hit in the head by RPG shrapnel," he said.
US troops encountered fierce
resistance in the city
"They hit us from one area and then another right afterwards. There is in-depth organisation. There were small-arms attacks all night," he said.
Journalist al-Badrani said US forces had taken some casualties.
He said two US tanks were destroyed in Julan neighbourhood, where the most violent clashes were taking place, and three armoured vehicles were destroyed in other parts of the city.
He added that fighters showed up from other neighbourhoods and streets that the US forces were unfamiliar with.
US forces entered central Falluja city but were fiercely attacked by fighters and withdrew from the area after half an hour, heading for their positions in the northern parts of the city.
Residents said the crews of two US tanks deserted their vehicles in Julan, leaving them to be seized by fighters.
The Pentagon said on Tuesday evening that at least 10 US and two Iraqi soldiers had died in the offensive unleashed by 10,000 US and 2000 Iraqi troops.
Up to 10,000 US troops are
participating in the offensive
US marines poured hundreds of rounds into rebel positions and blasted buildings with tank shells on Tuesday, but also took casualties, with bloodied troops stretchered away.
Explosions could be heard across Falluja after nightfall, but large-scale fighting appeared to have eased.
"I think we are looking at several more days of tough urban fighting," said the US commander in charge of day-to-day military operations in Iraq, Lieutenant-General Thomas Metz.
The assault on Falluja - where residents say wounded children are dying from lack of medical help, food shops are closed and power has been cut - angered Muslim clerics who urged Iraqis to boycott January elections seen as vital to peace.
Al-Badrani said many civilians had died in indiscriminate bombing of the city and people had resorted to burying their dead in gardens. Many houses have been destroyed.
Interim Prime Minister Iyad Allawi, who on Tuesday imposed a night curfew on Baghdad for an indefinite period, got a personal taste of the clerics' anger at a Ramadan iftar meal the same day.
"You have to stop fighting for four or five hours," Adnan al-Dulaimi, a Sunni official in the religious affairs ministry, told Allawi before the evening meal, a reporter said.
Iyad Allawi's (R) actions have
been criticised by Muslim clerics
"There are a lot of injured that have to be taken care of. Give them time to rescue the injured. There are civilians getting killed in Falluja. You are responsible for their lives in front of God," Dulaimi declared.
"As you know, we tried every alternative before resorting to military force," Allawi replied. "We have nothing against the civilians of Falluja... They are the sons of this country."
In a move that could potentially undermine the 27 January polls, the Sunni body, 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.
Residents say scores of civilians died and for those struggling to live in the city, life is grim.
Many of the city's 300,000 people had fled to escape air strikes and artillery bombardments preceding the assault. The US military said about half the residents had fled.
Those left behind say they have no power and use kerosene lamps. They keep to ground floors for safety, some living in shattered homes because it is too dangerous to move.