NEAR FALLUJAH, Iraq – American troops powered their way into the center of the terrorist stronghold of Fallujah yesterday, overwhelming small bands of guerrillas with massive force, searching homes along the city’s deserted, narrow passageways, and using loudspeakers to try to goad terrorists onto the streets.

As of last night, the fighting had killed 10 American troops and two members of the Iraqi security force, the American military announced. The toll already equaled the 10 American military deaths suffered when Marines besieged the city for three weeks in April.

American officials issued no estimate of insurgent casualties, but one American commander said his battalion alone had killed or wounded up to 90 terrorists.

As the offensive moved into a second full day, up to eight attack aircraft – including jets and helicopter gunships – blasted terrorist strongholds and raked the streets with rocket, cannon, and machine-gun fire ahead of American and Iraqi infantry who were advancing only one or two blocks behind the curtain of fire.

Small groups of terrorists, armed with rifles, rocket-propelled grenades, mortars, and machine guns, engaged American troops, then fell back. American troops inspected houses along Fallujah’s streets and ran across adjoining alleyways, mindful of snipers.

A psychological operations unit broadcast announcements in Arabic meant to draw out gunmen. An Iraqi translator from the group said through a loudspeaker: “Brave terrorists, I am waiting here for the brave terrorists. Come and kill us. Plant small bombs on roadsides. Attention, attention, terrorists of Fallujah.”

Faced with overwhelming force, resistance in Fallujah did not appear as fierce as expected, though the top American commander in Iraq said he still expected “several more days of tough urban fighting” as insurgents fell back toward the southern end of the city, perhaps for a last stand.

Some American military officers estimated they controlled about a third of the city. Commanders said they had not fully secured the northern half of Fallujah but were well on their way as American and Iraqi troops searched for terrorists.

American and Iraqi troops captured two key landmarks yesterday – a mosque and neighboring convention center that insurgents used for launching attacks, according to a Los Angeles Times reporter embedded with American forces.

“I’m surprised how quickly [resistance] broke and how quickly they ran away, a force of foreign fighters who were supposed to fight to the death,” Lieutenant Colonel Pete Newell, a battalion commander in the 1st Infantry Division, told CNN.

Mr. Newell was quoted on CNN’s Web site as saying his battalion had killed or wounded 85 to 90 insurgents.

The move against Fallujah prompted influential Sunni Muslim clerics to call for a boycott of national elections set for January. A widespread boycott among Sunnis could wreck the legitimacy of the elections, seen as vital in Iraq’s move to democracy. American commanders have said the Fallujah invasion is the centerpiece of an attempt to secure insurgent-held areas so voting can be held.

Prime Minister Allawi declared a nighttime curfew in Baghdad and its surroundings – the first in the capital for a year – to prevent terrorists from opening up a “second front” to try to draw American forces away from Fallujah.

Clashes erupted in the northern city of Mosul and near the Sunni bastion of Ramadi, explosions were reported in at least two cities, and masked terrorists brandished weapons and warned merchants to close their shops.

In Fallujah, American troops were advancing more rapidly than in April, when terrorists fought a force of fewer than 2,000 Marines to a standstill in a three-week siege. It ended with the Americans handing over the city to a local force, which lost control to Islamic terrorists.

This time, the American military has sent up to 15,000 American and Iraqi troops into the battle, backed by tanks, artillery, and attack aircraft. More than 24 hours after launching the main attack, American soldiers and Marines had punched through insurgent strongholds in the north and east of Fallujah and reached the major east-west highway that bisects the city.

“The enemy is fighting hard but not to the death,” Lieutenant General Thomas Metz, the multinational ground force commander in Iraq, told a Pentagon news conference relayed by video from Iraq. “There is not a sense that he is staying in particular places. He is continuing to fall back or he dies in those positions.”

General Metz said Iraqi soldiers searched several mosques yesterday and found “lots of munitions and weapons.”

Although capturing or killing the senior insurgent leadership is a goal of the operation, General Metz said he believed the most wanted man in Iraq, Abu Musab al-Zarqawi, had escaped Fallujah.

It was unclear how many terrorists stayed in the city for the fight, given months of warnings by American officials and Iraqis that a confrontation was in the offing.

General Metz said troops have captured a very small number of insurgent fighters and “imposed significant casualties against the enemy.”

Before the major ground assault that began Monday night, the American military reported 42 terrorists killed. Fallujah doctors reported 12 people dead. Since then, there has been no specific information on Iraqi death tolls.

The latest American deaths included two killed by mortars near Mosul and 11 others who died Monday, most of them as guerrillas launched a wave of attacks in Baghdad and southwest of Fallujah. It was unclear how many of those died in the Fallujah offensive, but the 11 deaths were among the highest for a single day since last spring.

But the toll in Fallujah could have been higher. Early yesterday, a helicopter gunship destroyed a multiple rocket launcher aimed at the main American camp outside of the city.

“That saved our lives,” Colonel Michael Formica, commander of the 1st Cavalry Division’s 2nd Brigade, told the crew. “We have no idea how many soldiers here were saved by your good work.”

American commanders said the operation was running on or ahead of schedule, and Iraqi officials designated an Iraqi general to run the city once resistance is broken.