US and Iraqi forces are fighting suspected insurgents house-to-house in a town near Iraq's border with Syria, in a major new offensive.
US troops have used Iraqi scouts to hunt down militants
About 3,500 US and Iraqi troops, backed by tanks and fighter aircraft, moved into Husayba two days ago.
A US military official said 36 insurgents and one US marine had been killed in the operation so far.
Medics working in the area said a similar number of civilians had died, and thousands had fled their homes.
Sunni Muslim politicians have criticised the operation, in the largely Sunni province of Anbar.
They say it will damage the chances of national reconciliation and cause more suffering to Iraqi people.
Ferdous al-Abadi, a spokesman for the Iraqi Red Crescent Society (IRCS), said the situation in and around Husayba was critical.
"People are really suffering," he told the United Nations news service.
The US military gave no information on civilian casualties but the IRCS said that 31 people had died and at least 44 had been injured.
There were reports that cluster bombs had been used, and that 4,000 people had fled the area. On Sunday the US said there were 450 civilian evacuees.
Meanwhile, a suicide car bombing in a southern district of Baghdad has killed nine people, including six Iraqi policeman.
Another 10 people were injured in the attack on a police convoy in the Dura district.
'I left everything'
The Husayba offensive, known as Operation Steel Curtain, is aimed at blocking what American and Iraqi officials say is a major infiltration route for militants crossing into Iraq.
About 200 suspected insurgents have been detained. The US military says its expects the death toll of suspected insurgents to rise once it is able to confirm numbers.
US troops have been ordered to find insurgents and their "safe houses" ahead of the Iraqi parliamentary election on 15 December, the US military has said.
In a statement on Sunday, the US military said the force was "clearing the city, house by house", taking fire from insurgents, and fighter jets had struck at least 10 targets around the town that day, the statement said.
The US military said it was not meeting "stiff resistance".
Civilians are said to have taken refuge in a vacant housing area in the town under the control of Iraqi forces.
"I left everything behind - my car, my house," one man said.
"I don't care if my house is bombed or looted, as long as I have my kids and wife safe with me."
The US military believes that the area around Husayba, about 320km (200 miles) west of Baghdad, is used by the al-Qaeda in Iraq group to smuggle in foreign fighters, money and equipment.
In a separate development, Iraqi President Jalal Talabani has urged Italy to keeps its 3,000 troops in Iraq, saying that to withdraw them prematurely would be a catastrophe for his country.
The Iraqi president spoke ahead of a visit to Italy which will include a meeting with Pope Benedict in the Vatican.