About a dozen US troops have died so far in the offensive against the city of Falluja, US Lieutenant General Thomas Metz said on Tuesday without giving a precise toll.
According to Pentagon figures, 10 soldiers were also wounded in and around Falluja.
"Friendly casualties are light," Metz said in a video teleconference briefing from Iraq.
The US military reported that two Iraqi soldiers of the interim army had been killed in the Falluja attack.
"Enemy casualties, I think, are significantly higher than I expected," Metz added, but declined to provide a number.
Mosul, Baghdad fighting
Three US military personnel were also reported killed in Mosul and Baghdad on Tuesday.
Fighting has increased in and
around the northern city of Mosul
"One Task Force Olympia soldier was killed and a second service member later died of wounds following a mortar attack on a multinational base in Mosul at approximately 10am (0700 GMT) today," a statement said.
"A civilian contractor also wounded in the attack was evacuated to the military hospital in Baghdad," it said.
Another US soldier succumbed to his wounds sustained during a firefight with Iraqi fighters in Baghdad.
The latest US deaths brings to 1140 the number of US troops killed in Iraq since the US-led invasion in March 2003, according to Pentagon figures.
In a separate development, anti-US fighters took control of the centre of Ramadi after 24 hours of clashes with US forces, an AFP correspondent has said.
The US military could not immediately be contacted for comment.
US forces withdrew on Tuesday at about 2pm (1100 GMT) from Ramadi's main streets to their bases east and west of the city, the correspondent said.
Five US troops were wounded in
clashes in Ramadi
Earlier, five US troops were wounded in Ramadi when marines shot at and destroyed two cars killing seven fighters, the US military said on Tuesday.
The attack occurred in the city on Monday, located 113km west of Baghdad, where US troops have clashed with fighters for weeks, the military said. No other details were available.
Ramadi, and Falluja to its east, are centres for anti-US fighters waging a 17-month campaign against US-backed interim authorities in Iraq.