US marines have taken the mayor's office in central Falluja and say they now control 70% of the Iraqi city, with rebels hemmed into a narrow strip.
Rebels set an oil pipeline on fire outside Falluja on Tuesday night
US officials say insurgents are now in "small pockets", moving blindly through the city, while fighting continues to rage for control of the centre.
According to US estimates, hundreds of rebels were killed on Wednesday alone and at least one more marine died.
The Red Cross has urged both sides to allow access for medical workers.
The BBC's Paul Wood, embedded with US marines who re-took the mayor's office, said that on Wednesday morning, no civilians could be seen on the streets, shops were shuttered and black smoke was rising all around.
Fadhil Badrani, a journalist in Falluja who reports for the BBC World Service in Arabic, compared the city to Kabul, the Afghan capital largely reduced to rubble after years of warfare.
One marine officer, Maj Francis Piccoli, said the rebels had been squeezed into a strip of the city bordering the main east-west road, which splits Falluja.
Another officer, Lt Gen John Sattler, said of the insurgents: "They are now in small pockets, blind, moving across the city. We will continue to hunt them down and destroy them."
Iraq's government has offered an amnesty to any armed groups in Falluja who surrender and are not found to have committed any "major" crimes.
In another development, the Arabic television station al-Jazeera has aired a videotape with Falluja militants displaying what they claim are 20 Iraqi soldiers taken captive.
Earlier, the chief spokesman for the joint US-Iraqi operation in Falluja, Maj Gen Abdul Qader Mohan, told reporters that Iraqi troops had found houses where hostages had been held and "slaughtered".
Elsewhere in Iraq on Wednesday:
- Militants abduct a first cousin of Iraqi interim Prime Minister Iyad Allawi and two of his family members in Baghdad, reportedly threatening to kill them unless the Falluja siege is lifted
- The governor in Mosul imposes an indefinite curfew after militants kill four members of the Iraqi security forces and a foreign contractor
- A car bomb in the residential area of Zayouneh in east Baghdad kills at least seven people
- Attacks at Balad, Baiji, Karbala and Tuz near Kirkuk leave 12 members of the Iraqi security forces and one US soldier dead.
Bodies in the streets
In Falluja, marines backed by tanks met little opposition when they blasted their way into the mayor's compound, which also houses a police station, early on Wednesday.
However, they later came under sustained fire from the minaret of a mosque, says our correspondent, whose reports are subject to military restrictions.
According to marines, the rebels waved a white flag at one stage but opened fire from three directions when a marine interpreter tried to begin talks. The marines then called in air strikes.
Fadhil Badrani told the BBC News website that the battle was particularly fierce in the district of Jolan, just north of the centre.
FALLUJA ASSAULT FACTS
Up to 15,000 US and Iraqi troops involved
Estimated 3,000 Iraqi insurgent and foreign fighters in city
Estimated 50,000 civilians remain out of usual population of some 300,000
He said he had counted the bodies of at least six US soldiers lying in the streets of the city's Hasbiyyah area overnight, along with the remains of many dead rebels. He had also found two disabled US tanks and three destroyed Humvee jeeps.
The journalist said he doubted the truth of US claims that marines were in control of 70% of the city.
The US military's total death toll for the Falluja operation rose to 11 Americans and two Iraqi government soldiers on Wednesday.
US commanders say rebel leaders such as Abu Musab al-Zarqawi - America's most wanted man in Iraq - appear to have fled before the assault began.
In Washington, President George W Bush praised the US-led forces in Falluja for their "hard work... for a free Iraq".
Aid agencies have highlighted the plight of civilians in Falluja where up to 50,000 people remain out of a pre-war population of 300,000.
The Red Cross has urged all combatants to guarantee passage to the wounded.
An unnamed man claiming to be a rebel fighter told the BBC's Today programme that the destruction in Falluja was "total".
"The Americans are bombing everywhere," he said, adding that water and electricity had been cut off.
The assault on Falluja, a hotbed of Sunni resistance, is officially aimed at stabilising Iraq ahead of January's poll.