U.S. forces pushed into the heart of Fallujah on Tuesday, encountering roadside bombs, rockets and gunfire on the second day of a battle to wrest control of the city from insurgents.

Army and Marine units that entered Fallujah from the northeast and northwest on Monday night had fought their way to the city center and beyond by Tuesday night, U.S. commanders said.

Soldiers with the Army's 1st Infantry Division made their way to the southeastern part of the city, a neighborhood of factories and warehouses where they expected to find guerrillas waiting for them. Instead, the district was relatively quiet, though the units reported being fired on by women and children armed with assault rifles.

"There were multiple groups running around shooting at us," said Air Force Senior Airman Michael Smyre, 26, of Hickory, N.C., an airstrike spotter attached to the 1st Infantry who was wounded when a rocket hit his armored vehicle. "You could see a lot of rubble, trash everywhere. It was real nasty-looking."

Marines fighting to the west of the Army units advanced to the main east-west highway that divides Fallujah and reported persistent resistance from insurgents firing from mosques.

The U.S. military said 10 troops and two members of Iraq's security forces were killed in the first two days of the battle, the largest military operation since the U.S.-led invasion last year. U.S. and Iraqi leaders hope the assault will break the grip of insurgents who have held Fallujah for nearly seven months.

Some Iraqi political and religious groups condemned the push into Fallujah, a stronghold of the Sunni Muslim minority. A leading Sunni organization, the Iraqi Islamic Party, quit the country's interim government, and Sunni clerics on Tuesday made good on threats to call for a boycott of January elections. Harith Dhari, head of the pro-insurgency Association of Muslim Scholars, said balloting would occur "over the corpses of those killed in Fallujah."

Insurgents elsewhere in Iraq, meanwhile, continued a strategy of mounting attacks. In Baqubah, a restive city northeast of Baghdad, armed bands attacked two police stations. Police officials and the U.S. military said the attacks were beaten back. A car bomb at an Iraqi National Guard camp outside the northern city of Kirkuk killed three people and wounded two. And two U.S. service members were killed in a mortar attack on a base in Mosul, also in the north.

In Baghdad, where insurgents on Monday night detonated a car bomb outside a hospital treating victims of two car bombs outside churches, interim Prime Minister Ayad Allawi imposed a curfew from 10:30 p.m. to 4 a.m. U.S. fighter jets made low passes over the capital, a show of strength rarely seen since the 2003 invasion.

At a news conference in Baghdad, Lt. Gen. Thomas Metz, the commander of foreign military operations in Iraq, said the assault on Fallujah had so far "achieved our objectives on or ahead of schedule." He added, "I think we're looking at several more days of tough urban fighting."

The general said the battle plan as a whole was on course. "We felt like the enemy would form an outer crust in defense of Fallujah. We broke through that pretty quickly and easily," Metz said. "We also then anticipated him breaking up into small three- to six-person detachments or squads, which we've seen throughout the day, today especially."

Witnesses said that by Tuesday night, U.S. and Iraqi forces controlled the Jolan, Mualimeen and Askali neighborhoods in the north of Fallujah. They also held the Rawdha Muhammediya mosque, headquarters of the insurgent fighters and the mujaheddin shura, the city's self-appointed government.

The assault pushed insurgents into Shuhada and other neighborhoods in the southernmost part of the city, where they are fighting and hiding behind buildings and houses, witnesses said.

Metz said that because U.S. forces formed a "very tight" cordon around the city Sunday night, the enemy "doesn't have an escape route" and eventually would be cornered.

But Sheik Abdul-Sattar Edatha, the spokesman for the shura council, said most foreign fighters had already left the city. The U.S. military had estimated that there were 2,000 to 3,000 foreign fighters in the city, many of them part of a network linked to Jordanian-born guerrilla leader Abu Musab Zarqawi.

"Militarily speaking, the city falls under the U.S. forces' control," Edatha said. "The foreign fighters won't stay here and die. They lost the battle. They spread in other places."

On Tuesday night, Fallujah's eerily empty streets were littered with shattered concrete and dead bodies, said a resident shaken by a missile strike on the second story of his family home. Insurgents cloaked in checkered head scarves carried wounded fellow fighters to mosques.

Civilians caught in the crossfire were gathered in a hospital donated by the United Arab Emirates and flying a blue and white UNICEF banner. There, medical workers low on bandages and antiseptic bound wounds in ripped sheets and cleaned torn skin with hot water.

The Jolan and Askali neighborhoods seemed particularly hard hit, with more than half of the houses destroyed. Dead bodies were scattered on the streets and narrow alleys of Jolan, one of Fallujah's oldest neighborhoods. Blood and flesh were splattered on the walls of some of the houses, witnesses said, and the streets were full of holes.

Some of the heaviest damage apparently was incurred Monday night from air and artillery attacks that coincided with the entry of ground troops into the city. U.S. warplanes dropped eight 2,000-pound bombs on the city overnight, and artillery boomed throughout the night and into the morning.

"Usually we keep the gloves on," said Army Capt. Erik Krivda, of Gaithersburg, the senior officer in charge of the 1st Infantry Division's Task Force 2-2 tactical operations command center. "For this operation, we took the gloves off."

Some artillery guns fired white phosphorous rounds that create a screen of fire that cannot be extinguished with water. Insurgents reported being attacked with a substance that melted their skin, a reaction consistent with white phosphorous burns.

Kamal Hadeethi, a physician at a regional hospital, said, "The corpses of the mujaheddin which we received were burned, and some corpses were melted."

In addition to ripping open entire neighborhoods, the armor assault also brought into the open an insurgent command that until this week remained shadowy even to Fallujah residents. Ex-generals from the former Iraqi army's Republican Guard passed written orders, complete with official stamp, to subordinates who snapped salutes, witnesses said.

Iraq's new army, formed after occupation authorities dismantled the armed forces that had served during the rule of Saddam Hussein, is taking part in the fight against insurgents in Fallujah, primarily as a rear element to help clear areas once U.S. forces have moved through. Marine commanders have declined to comment on the offensive, deferring to Iraqi officers. On Tuesday, Brig. Gen. Abdul-Qadir Muhammed Jasim characterized the offensive as "a holy task to fight for Fallujah people."

"We will fight to the last drop of our blood to free our people," he said at a news conference just outside the city. "We will fulfill the tasks we've been asked to do, with the cooperation of our friends."

Jasim said that resistance had been lighter than expected and that the Iraqi soldiers were in good spirits and eager to finish the operation.

"The operation is going very precise and with a very small number of casualties," he said. "In every place we finish an operation, our forces start to distribute aid, food, clothes, blankets and even money. . . . We are very sure that we are moving in the right way and will do the tasks we are asked to do very precisely."

Metz repeatedly praised Iraqi forces, saying they had "acquitted themselves very well in this fight." Metz said the Iraqi soldiers had been used especially to search the city's 77 mosques. "In several mosques today, lots of munitions and weapons were found, and they were found by those Iraqi soldiers," he said.

Metz's account suggested a marked improvement among the Iraqi troops in recent months. In April, the last time U.S. commanders tried to use Iraqi forces in Fallujah, a battalion of freshly trained Iraqi troops refused to go.

A senior Iraqi official said it was too early to tell how the Iraqi forces performed. "During the operation you always hear they're doing good," said Industry Minister Hachim Hasani. "After the operations are finished, we'll find out."

Hasani's political organization, the Iraqi Islamic Party, quit the interim government Tuesday to protest the Fallujah offensive. But Hasani, who opposed the U.S. Marine siege of the city earlier this year, quit the party Tuesday and retained his cabinet post. "Iraq is larger than any party," Hasani said. "Things should be done through the government, not outside the government."

Vick and special correspondent Bassam Sebti reported from Baghdad.

Marines in Fallujah advanced to the city's main east-west highway and reported insurgents firing from mosques. Ten U.S. and two Iraqi troops have died in two days of battle.Marines from Charlie Company of the 1st Battalion, 8th Regiment take up positions along a narrow street in Fallujah.