Some 50 rebels have been killed and another 100 captured since the start of a joint US-Iraqi offensive in western Iraq, the US military says.
About 2,000 US and Iraqi troops are taking part in the two operations
It reported no military or civilian losses since Operation Spear began on Friday, but Iraqi doctors spoke of at least 10 civilian deaths.
The operation - involving some 1,000 troops - aims to root out rebels and foreign fighters on the Syrian border.
A second major operation in the area began on Saturday to find arms caches.
About 1,000 US and Iraqi troops were taking part in Operation Dagger, which also targeted what the US military believed was a rebel base north-west of the capital, Baghdad.
The two operations follow a week of attacks which left at least 11 American soldiers dead.
"Approximately 50 insurgents have been killed since the operation began yesterday [Friday] morning," US marine Capt Jeffrey Pool said.
The offensive in the province of Anbar began before dawn on Friday, with witnesses reporting fierce gun battles in the towns of Karabila and Qaim.
Fighter planes dropped bombs on suspected rebel safe houses and weapons caches in the area, some 20km (12 miles) east of Iraq's border with Syria.
Other bombs reportedly targeted rebels who were firing at US ground forces.
"Marines and Iraqi soldiers continued operations through the night securing key objectives in and around the city [of Karabila]," Capt Pool said.
He said four Iraqi hostages had been found beaten, handcuffed and chained to a wall in a bunker located in central Karabila.
Meanwhile, Iraqi doctors in Qaim said they had received the bodies of at least 10 Iraqi civilians.
American officials say the western border area has become a new haven for insurgents who sheltered in Falluja until the US offensive last November.
They say that foreign fighters continue to slip into the country from Syria, but Damascus rejects the accusations.