U.S. and Iraqi forces pushed deeper into Fallouja today, taking control of 70% of the city, including mosques, the City Hall complex and other key buildings as they searched house to house for weapons and guerrillas.

Also today, gunmen kidnapped two members of interim Prime Minister Ayad Allawi's family in Baghdad. His cousin, Ghazi Allawi, and the cousin's daughter-in-law were abducted Tuesday night.

Troops from Alpha Company, 1st Battalion, 8th Marine Regiment, seized the City Hall near the city center without major resistance this morning as troops began the third day of their major offensive to take control of the insurgent stronghold, Marines said. Heavy fighting continued in some areas, including the Jolan district.

Several units encountered heavy fire from snipers and squads of guerrillas, but U.S. military leaders said that overall, resistance was lighter than expected and the advance was proceeding more quickly than anticipated.

Maj. Francis Piccoli of the 1st Marine Expeditionary Force said U.S. and Iraqi forces controlled 70% of the city after cornering insurgents into a small section of Fallouja. "The heart of the city is what's in focus now," he was quoted as saying by the Associated Press.

At least 10 U.S. troops and two Iraqi soldiers had been killed in the operation. The U.S. gave no overall tally of civilian or insurgent casualties, but Army Lt. Col. Pete Newell, a battalion commander with the 1st Infantry Division, told CNN that his unit had killed or wounded at least 85 guerrilla fighters.

In Washington, President Bush met Wednesday with Army Gen. George W. Casey, the top U.S. military commander in Iraq, to review the status of the three-day offensive.

"He said that things are going well in Fallouja, and they're making very good progress in securing that country," Bush told reporters following a White House meeting with NATO Secretary General Yap de Hoop Sheffer.

Bush said he was willing to increase the number of U.S. troops in Iraq if requested by military authorities there, but he said no such request had been made. "If the commanders were to bring forth a request, I would look at it. I would listen to it very seriously, and implement the request. They have yet to do so," Bush said.

Elsewhere in Iraq, more than a dozen others have died in violence, including a U.S. soldier who was killed today in Tikrit. Pitched battles took place in Baghdad, in the northern city of Mosul, and in Ramadi to the south.

Lt. Gen. Thomas F. Metz, the top operational commander in Iraq, said Tuesday that "several more days of tough urban fighting" would be needed for U.S. forces to complete their sweep through Fallouja.

Metz said militants were fighting in small squads of three to six and did not appear to have a comprehensive plan to defend the city.

"I think the enemy is fighting hard, but not to the death, and I think that they are continuing to fall back," he said, speaking via teleconference from Iraq to reporters at the Pentagon.

Before the assault began, American officials estimated that 3,000 to 5,000 insurgents were in Fallouja. U.S. commanders were unsure Tuesday whether the lighter-than-predicted fighting was a sign that fewer were left in the city than estimated, or whether they might be retreating toward the city center to stage a coordinated defense.

Even as commanders said the Fallouja offensive was progressing according to plan, violence flared in other areas of Iraq. Two U.S. soldiers were killed in a mortar attack in the northern city of Mosul, and insurgents attacked two police stations in Baqubah, injuring eight Iraqi policemen, the military said.

And in a statement that appeared on an Islamist website, insurgent groups warned Iraqis in Baghdad and other cities to remain at home today to avoid "putting their lives in danger."

The warning posted on the Internet was also distributed on leaflets early today in neighborhoods of Baghdad where the insurgency is active. It specifically said government workers should stay away from their offices, with the exception of those at the Health Ministry. The leaflet was signed by 10 insurgent groups, most of which had claimed responsibility for previous attacks on Iraqi government forces and Westerners.

Allawi imposed a nighttime curfew in Baghdad on Tuesday, the first in more than a year. In Fallouja, a group calling itself the Ansar Jihad threatened to behead his family members within 48 hours unless the offensive was called off, but their claims of holding the kidnapping victims have been unsubstantiated.

The offensive to retake Fallouja, which has been controlled by insurgents since April, is part of a strategy by the U.S. military and Allawi's administration to stabilize the country and reduce violence ahead of elections for a national assembly planned Jan. 31.