U.S. forces stormed the Sunni guerrilla-controlled city of Fallujah (search) on Monday in a massive air and ground offensive.

A military spokesman estimated that about 42 of the rebels who have held the city near Baghdad since the spring were killed before the major attack even started. Two U.S. Marines died when their bulldozer dropped into the Euphrates River near Fallujah.

Backed by thunderous air and artillery barrages, American troops launched the extensive assault Monday morning with the goal of seizing key insurgent strongholds inside the city ahead of January elections in Iraq.

Fallujah became Iraq's major sanctuary for Islamic extremists, who fought Marines to a standstill last April and in one of the most grim images of the war to date, strung the badly burned bodies of four American soldiers from a bridge there.

Hours after starting the offensive, U.S. tanks and Humvees from the 1st Infantry Division entered the northeastern Askari neighborhood, the first ground assault into an insurgent bastion.

In the northwestern area of the city, U.S. troops advanced slowly after dusk on the Jolan neighborhood, a warren of alleyways where Sunni (search) militants have dug in. Artillery, tanks and warplanes pounded the district's northern edge, softening the defenses and trying to set off any bombs or boobytraps planted by the militants.

Marines were visible on rooftops in Jolan. An Associated Press reporter, located at a U.S. camp near the city, saw orange explosions lighting up the district's palm trees, minarets and dusty roofs, and a fire burning on the city's edge.

FOX News' Greg Palkot, embedded with the 1st Marine Expeditionary Force, said Monday that the unit was overlooking the northwest corner of Fallujah, where a series of tanks had been pounding positions inside the city for two hours.

Ground commanders targeted three areas they believed guerrillas were operating and it looked as if preparations were being made for a full-scale ground assault, Palkot reported.

Masked insurgents roamed Fallujah streets throughout the day. One group of four fighters, two of them draped with belts of ammunition, moved through narrow passageways, firing on U.S. forces with small arms and mortars. Mosque loudspeakers blared, "God is great, God is great."

Just outside the Jolan and Askari neighborhoods, Iraqi troops deployed with U.S. forces took over a train station after the Americans fired on it to drive off fighters.

The top U.S. commander in Iraq, Gen. George Casey (search), predicted a "major confrontation" in the operation he said was called "al-Fajr" (search), Arabic for "dawn." He told reporters in Washington that 10,000 to 15,000 U.S. troops along with a smaller number of Iraqi forces were encircling the city.

Overall, the main force did not appear to have moved deeply into Fallujah on Monday, the first full day of the operation. Most U.S. units appeared to be lined up at the edge of their neighborhoods with some scouts and perhaps special operators venturing inside.

The offensive is considered the most important military effort to re-establish government control over Sunni strongholds west of Baghdad before elections in January.

"One part of the country cannot remain under the rule of assassins ... and the remnants of Saddam Hussein's regime," Defense Secretary Donald H. Rumsfeld (search) said. He predicted "there aren't going be large numbers of civilians killed and certainly not by U.S. forces."

A doctor at a clinic in Fallujah, Mohammed Amer, reported that 12 people were killed. Seventeen others, including a 5-year-old girl and a 10-year-old boy, were wounded, he said.

About 3,000 insurgents were barricaded in Fallujah, U.S. commanders have estimated. Casey said some insurgents slipped away but others "have moved in." U.S. military officials think that 20 percent of Fallujah's fighters are foreigners, who are believed to be followers of Jordanian militant Abu Musab al-Zarqawi (search).

Casey said 50 to 70 percent of the city's 200,000 residents have fled. The numbers are in dispute, however, with some putting the population at 300,000. Residents said about half that number left in October, but many drifted back.

Some 5,000 U.S. Marines and soldiers were massed in the desert on Fallujah's northern edge. They were joined by 2,000 to 4,000 Iraqi troops.

Rumsfeld called reports of some Iraqi recruits not showing up to fight "an isolated problem," and Casey said the no-shows "did not have a significant impact" on the operation.

Iraqi Prime Minister Ayad Allawi (search), who gave the green light for the offensive, also announced a round-the-clock curfew in Fallujah and another nearby insurgent stronghold, Ramadi.

"The people of Fallujah have been taken hostage ... and you need to free them from their grip," he told Iraqi soldiers who swarmed around him during a visit to the main U.S. base outside Fallujah.

"May they go to hell!" the soldiers shouted, and Allawi replied: "To hell they will go."

U.S. commanders have avoided any public estimate on how long it may take to capture Fallujah, where insurgents fought the Marines to a standstill last April in a three-week siege.

The length and ferocity of the battle depends greatly on whether the bulk of the defenders decide to risk the destruction of the city or try to slip away in the face of overwhelming force. Foreign fighters may choose to fight to the end, but it's unclear how many of them are in the city.

Rumsfeld said insurgents would likely put up a tough fight. "Listen, these folks are determined. These are killers. They chop people's heads off. They're getting money from around the world. They're getting recruits," he told reporters.

But the Iraqi defense minister, Hazem Shaalan al-Khuzaei (search), told Al-Arabiya television that he expected the resistance to crumble quickly.

"God willing, it will not be long; it will take a very short period of time," he said, adding that the insurgents might use the civilians as human shields.

As the main assault began in Fallujah, thunderous explosions could be heard across Baghdad, some 40 miles to the east. Militants detonated car bombs in quick succession near two churches after sundown, killing six people and injuring 52 others.

A U.S. soldier was killed when his patrol was fired on in Baghdad, the military said. Southwest of the capital, a British soldier died in an apparent roadside bombing.

The prelude to the Fallujah ground offensive was a crushing air and artillery bombardment that built from the night before, through Monday morning and afternoon, then rose to a crescendo by Monday night — with U.S. jets dropping bombs constantly and big guns pounding the city every few minutes with high-explosive shells.

Associated Press reporter Edward Harris, embedded with the Marines near the train station in the desert north of the city, saw U.S. forces hammering Jolan with airstrikes and intense tank fire. The Marines reported that at least initially they did not draw significant fire from insurgents, only a few rocket-propelled grenades that caused no casualties.

Earlier Monday, U.S. and Iraqi forces seized two bridges over the Euphrates River and a hospital on Fallujah's western edge that they said was under insurgents' control. A team of Marines entered northwestern Fallujah and seized an apartment building.

Capt. Jonathan Riley, spokesman for the U.S. Central Command Air Forces in Qatar, told the AP that an unmanned MQ-1 Predator plane fired a Hellfire missile at an insurgents' anti-aircraft artillery battery in Fallujah, scoring a direct hit.

The Association of Muslim Scholars, a Sunni clerics group that has threatened to boycott elections, condemned the assault on Fallujah, calling it "an illegal and illegitimate action against civilian and innocent people."

Arab leaders were muted in their response to the offensive. Media attention focused on ailing Palestinian leader Yasser Arafat, which may explain in part why the start of the Fallujah campaign elicited none of the uproar that met the American attempt to storm the insurgent stronghold in April.

The Associated Press contributed to this report.