Updated

At least 12 U.S. Marines were killed Tuesday in an "extremely intense" battle in Ramadi (search), another Sunni Triangle hotbed of guerrilla activity 24 miles west of Fallujah, Pentagon and military officials told Fox News on Tuesday.

Reports from the city of Ramadi said dozens of Iraqis attacked a Marine position near the governor's palace, a senior defense official said from Washington.

It was initially thought the fighters were in the army of Muqtada al-Sadr (search), the fiery anti-American Shiite Muslim cleric wanted on a murder charge. However, senior U.S. officials said there is a growing feeling that most of the attackers were former regime elements and Baath Party members and not just members of Shiite militias.

A defense official told Fox News that there have been "significant" U.S. deaths from the fighting in Ramadi — the official later set the figure at at least 12 U.S. casualties and at least 20 wounded.

Two more coalition soldiers and at least 66 Iraqis were also reported killed Tuesday.

President Bush, in a statement released from the White House Tuesday night, said, "Our troops are performing well."

Fighting was so intense that commanders went to the unit headquarters to pull people who wouldn't normally fight into the combat, military sources said.

"This is not like any other firefight we've seen so far," military sources in Iraq said. "There are bullets flying all over the place."

A U.S. official in Washington said all American officials in Iraq, including those working for the Coalition Provisional Authority (search), have been told since Monday to remain inside their compounds due to security worries.

Meanwhile, al-Sadr, sought by U.S. and Iraqi forces, announced Tuesday that he had left a mosque in the city of Kufa, south of Baghdad, where he has been holed up for the past two days, surrounded by armed militiamen.

U.S. administrators on Monday declared al-Sadr an "outlaw" and announced a warrant for his arrest. He was demanding to speak to high-level coalition officials Tuesday and said he would not negotiate his surrender.

Rumsfeld Backs More Troops

Insurgents and rebellious Shiites mounted a string of attacks across Iraq's Shiite south as U.S. Marines launched a major assault on the turbulent Sunni city of Fallujah — all part of some of the most extensive fighting since President Bush declared major combat over on May 1.

With fighting intensifying ahead of the June 30 handover of power to an Iraqi government, Secretary of Defense Donald H. Rumsfeld said American commanders in Iraq would get additional troops if needed. None has asked so far, he said.

"They will decide what they need, and they will get what they need," Rumsfeld said.

Depending on the number of Ramadi deaths, Tuesday's casualties could have brought the three-day total as high as about 30 Americans and more than 130 Iraqis killed in the fighting.

Operation Vigilant Resolve Moves Into Fallujah

On the Fallujah front, Marines drove into the center of the Sunni city in heavy fighting before pulling back before nightfall as part of Operation Vigilant Resolve (search). The assault had been promised after the brutal killings and mutilations of four American civilians there last week. Hospital officials said eight Iraqis died Tuesday and 20 were wounded, including women and children.

Marines waged a fierce battle for hours Tuesday with gunmen holed up in a residential neighborhood of Fallujah. The military used a deadly AC-130 gunship to lay down a barrage of fire against guerrillas, and commanders said Marines were holding an area several blocks deep inside the city. At least two Marines were wounded.

U.S. warplanes firing rockets destroyed four houses in Fallujah after nightfall Tuesday, witnesses said. A doctor said 26 Iraqis, including women and children, were killed and 30 wounded in the strike. The deaths brought to 34 the number of Iraqis killed in Fallujah on Tuesday.

The dusty, Euphrates River city 35 miles west of Baghdad is a stronghold of the anti-U.S. insurgency that sprang up shortly after Saddam Hussein's ouster a year ago.

"We are several blocks deep in the city of Fallujah," Marine Maj. Briandon McGolwan said. He said several helicopters were hit by small arms fire, but none were downed. He said Marines had detained 14 people since Monday.

U.S. Marines encircled Fallujah early Monday, and on Tuesday, they penetrated several central neighborhoods for the first time. Mortar and rocket-propelled grenade blasts were heard, and one witness said a Humvee was ablaze.

Heavy fighting also occurred between Marines entrenched in the desert and guerrillas firing from houses on Fallujah's northeast outskirts. For hours into the night, the sides traded fire, while teams of Marines moved in and out of the neighborhood, seizing buildings to use as posts and battling gunmen. Helicopters weaved overhead, firing at guerrilla hide-outs.

U.S. authorities launched their offensive against al-Sadr and his militia after a series of weekend uprisings in Baghdad and cities and towns to the south that took a heavy toll in both American and Iraqi lives.

Battles Rage Across Iraq

The fight against al-Sadr, who has drawn backing from young and impoverished Shiites with rousing sermons demanding a U.S. withdrawal, sent his black-garbed militiamen against coalition troops Sunday, Monday and Tuesday.

Fighting in the southern cities of Nasiriyah, Kut, Karbala and Amarah and in a northern Baghdad neighborhood killed 32 Iraqis, coalition military officials said. Tuesday evening, gunfire was heard in another part of Baghdad, Sadr City, where fierce battles occurred Sunday, residents said.

— In Nasiriyah on Tuesday, 15 Iraqis were killed and 35 wounded in clashes between militiamen and Italian troops, coalition spokeswoman Paola Della Casa told an Italian news agency Apcom. Eleven Italians troops were slightly wounded.

Della Casa said the Iraqi attackers used civilians as human shields, and a woman and two children were among the dead.

— Fighting overnight in Amarah between al-Sadr's followers and British troops killed 15 Iraqis and wounded eight, said coalition spokesman Wun Hornbyckle.

— In Kut, militiamen attacked an armored personnel carrier carrying Ukrainian soldiers, killing one and wounding five, the Ukrainian Defense Ministry said. Two militiamen were killed in the fight. Ukraine has about 1,650 troops in Iraq.

L. Paul Bremer, the top civilian administrator in Iraq, conceded not all was going smoothly as the coalition approached the June 30 handover, a date he said was inviolable.

"The political process is well under way with broad consultations with the Iraqi people. The interim government will be in place well before June 30," the date to hand over government operations to the Iraqis, Bremer told Fox News.

'America Has Shown Its Evil Intentions'

Fearing a U.S. move to arrest him, al-Sadr on Tuesday left the fortress-like mosque in the city of Kufa.

"America has shown its evil intentions, and the proud Iraqi people cannot accept it," al-Sadr said in a statement. "They must defend their rights by any means they see fit."

The statement said he was ready to die to oust the Americans, and he urged his followers to resist foreign forces.

Al-Sadr moved to his main office in Najaf, in an alley near the city's holiest shrine, according to a top aide, Sheik Qays al-Khaz'ali. Hundreds of militiamen were protecting the office Tuesday, but there was no independent confirmation al-Sadr was there.

State Department deputy spokesman Adam Ereli said al-Sadr and his followers were not representative of a religious cause but of "political gangsterism."

The 30-year-old al-Sadr, however, does not have a large following among majority Shiites — many see him as a renegade, too young and too headstrong to lead wisely.

Perhaps more worrisome than the current fight with al-Sadr's forces is the possibility that he will start drawing support from more mainstream Shiite leaders who have largely supported the Americans until now.

The U.S.-led coalition announced a murder warrant against al-Sadr on Monday and suggested it would move to capture him soon. It said an unnamed Iraqi judge had issued it in the past months.

The militiamen clashed with coalition troops Sunday in Baghdad and outside Najaf in fierce fighting that killed 61 people, including eight American soldiers.

A U.S. soldier was killed in Baghdad Tuesday, a day after two more were killed there. Five Marines were killed Monday — one in Fallujah and the others on the western outskirts of Baghdad. On Sunday, two soldiers were killed in Kirkuk and Mosul.

Excluding the report out of Ramadi on Tuesday evening, at least 614 American troops have died in Iraq since the war began.

Fox News' Bret Baier, Steve Centanni and The Associated Press contributed to this report.