American forces bottled up guerrillas in a narrow strip of Fallujah's (search)s alleys and streets Wednesday after a stunningly swift advance that seized control of 70 percent of the insurgent stronghold. In Baghdad, kidnappers abducted two members of Prime Minister Ayad Allawi's (search) family, the government said.

A militant group calling itself Ansar al-Jihad (search) claimed in a Web posting to have carried out the kidnapping and threatened to behead the hostages within 48 hours unless the siege of Fallujah was lifted and prisoners were freed. The claim's authenticity could not immediately be verified.

Armed men snatched one of the prime minister's cousins, Ghazi Allawi, and the cousin's daughter-in-law from their home in Baghdad's western Yarmouk neighborhood Tuesday night, government spokesman Thair al-Naqeeb said. "Ghazi Allawi is 75 years old. He has no political affiliation, and is not holding a government post," al-Naqeeb said.

Ansar al-Jihad said in its Web posting that it abducted three people — a cousin of Allawi, the cousin's wife and another relative. "We promise Allah and his messenger that if the agent government doesn't respond to our demands within 48 hours, they (the hostages) will be beheaded," the statement said. Police had initially said that three relatives were kidnapped.

Insurgents have been trying to open a "second front" with a wave of attacks to divert U.S. forces from their offensive in Fallujah.

In Fallujah, the military said U.S. troops pushed insurgents into a section of the city flanking the main east-west highway that bisects the rebel bastion. At least 71 militants had been killed as of the beginning of the third day of intense urban combat, the military said, with the casualty figure expected to rise sharply once U.S. forces account for Iraqis and foreign fighters killed in airstrikes.

As of Tuesday night, 10 U.S. troops and two members of the Iraqi security force had been killed, a toll that already equaled the number of American troops who died when Marines besieged the city for three weeks in April.

Major Francis Piccoli, of the 1st Marine Expeditionary Force, characterized fighting overnight as "light to moderate" and said U.S. casualties were "extremely light."

Piccoli said U.S. forces that pushed south through Fallujah's central highway overnight now control 70 percent of the city. He said troops would move on Wednesday into the strip of territory where guerrillas were bottled up. "The heart of the city is what's in focus now," he said.

The northwestern neighborhood of Jolan, the historic warren of crooked streets where Sunni militants and foreign fighters had rigged boobytraps, was now "secured and under control," he said, although Marines were expected to continue house-to-house searches for fighters and weapons.

About 100 men, women and children left their homes in Fallujah and made their way to American positions in the south of the city where they gave themselves up Wednesday, an officer from the Army's 1st Cavalry Division said. The group was to be searched for weapons and questioned, and all military-age men would be detained, the officer said.

Most of Fallujah's 200,000 to 300,000 residents are believed to have fled the city before the U.S. assault. Civilian casualties in the attack are not known, though U.S. commanders say they believe they are low. Officers on Tuesday said few civilians have tried to leave during the fighting, though a funeral procession was allowed to exit the city.

Marine reports Wednesday said 25 American troops and 16 Iraqi soldiers were wounded.

Also Wednesday, one U.S. soldier was killed and a second was wounded by a roadside bomb north of Baghdad. In northern Iraq, six Iraqi soldiers died and two were wounded when a roadside bomb detonated near an Iraqi military camp.

Guerrillas trying to open a "second front" have launched a wave of attacks that have killed at least 13 Americans since Monday — apart from the Fallujah operation. For the first time in a year, Allawi declared a nighttime curfew in Baghdad and its surroundings.

U.S. troops in Fallujah advanced a block or two behind a curtain of firepower laid down by as many as eight attack aircraft — including jets and helicopter gunships — which blasted guerrilla positions and raked the streets with rocket, cannon and machine-gun fire.

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

Despite resistance being lighter than expected, the top U.S. commander in Iraq said Tuesday he still predicted "several more days of tough urban fighting" as insurgents fell back toward the southern end of the city, perhaps for a last stand.

"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," Lt. Col. Pete Newell, a battalion commander in the 1st Infantry Division, told CNN.

The U.S. advance in Fallujah was more rapid than in April, when insurgents 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 militants.

This time, the U.S. military has sent up to 15,000 U.S. and Iraqi troops into the battle, backed by tanks, artillery and attack aircraft.

"The enemy is fighting hard but not to the death," Lt. Gen. 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."

Metz said Iraqi soldiers searched several mosques Tuesday and found "lots of munitions and weapons."

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

It was unclear how many insurgents stayed in the city for the fight. 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 U.S. military reported 42 insurgents killed. Fallujah doctors reported 12 people dead. Since then, there has been no specific information on Iraqi death tolls.