NEAR FALLUJAH, Iraq – American troops backed by thunderous air and artillery barrages launched a ground offensive yesterday to seize key insurgent strongholds inside Fallujah, the city that became Iraq’s major sanctuary for Islamic terrorists who fought Marines to a standstill last April.

Two Marines were killed when their bulldozer flipped over into the Euphrates near Fallujah, and a military spokesman estimated 42 terrorists were killed across the city in a bombardment and skirmishes before the main assault began.

Hours after starting the offensive, American tanks and Humvees from the 1st Infantry Division entered the northeastern Askari neighborhood, the first ground assault into a terrorist bastion.

In the northwestern area of the city, American troops advanced slowly after dusk on the Jolan neighborhood, a warren of alleyways where Sunni terrorists have dug in. Artillery, tanks, and warplanes pounded the district’s northern edge, softening the defenses and trying to set off any bombs or booby traps planted by the terrorists.

Marines were visible on rooftops in Jolan. This reporter, located at an American 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.

Heavy firing continued into the predawn hours today, and residents reached by satellite telephone reported the constant drone of warplanes overhead.

American troops cut off electricity to the city, and most private generators were not working. Residents said they were without running water and were worried about food shortages because most shops in the city have been closed for the past two days.

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 American 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 American forces took over a train station after the Americans fired on it to drive off fighters.

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

An American jet fired an air-to-ground missile at a building late yesterday from which American and Iraqi forces had taken fire, the American command said. The building was destroyed.

Overall, the main force did not appear to have moved deeply into Fallujah yesterday, the first full day of the operation. Most American units appeared to be lined up at the edge of the 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 Rumsfeld said. He predicted that “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 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 terrorists were barricaded in Fallujah, American commanders have estimated. General Casey said some insurgents slipped away but others “have moved in.” American military officials believe 20% of Fallujah’s fighters are foreigners, believed to be followers of Jordanian terrorist Abu Musab al-Zarqawi.

General Casey said 50% to 70% 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 American Marines and soldiers were massed in the desert on Fallujah’s northern edge. They were joined by 2,000 to 4,000 Iraqi troops.

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

Prime Minister Allawi, who gave the green light for the offensive, also announced a round-the-clock curfew in Fallujah and another nearby terrorist 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 American base outside Fallujah. “May they go to hell!” the soldiers shouted, and Mr. Allawi replied: “To hell they will go.”

American 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 un clear how many of them are in the city.

Mr. Rumsfeld said terrorists 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, told Al-Arabiya TV 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 terrorists 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. Terrorists detonated car bombs in quick succession near two churches after sundown, killing six people and injuring 52 others. An American 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 yesterday, then rose to a crescendo last night.