They also faced violence in the capital, where a suicide bomber targeted an army recruiting centre, killing 23 people.
The UN and NGOs have warned that civilians lack access to key supplies as the government blockades Fallujah and parts of the nearby provincial capital Ramadi, which were seized by militants last week.
And Washington has piled pressure on Baghdad to focus on political reconciliation as well as military operations to resolve the standoff.
Today, security forces engaged in heavy fighting with al-Qaeda-linked militants in the Albubali area, between Ramadi and Fallujah, a police officer said.
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Eventually, tank and helicopter fire destroyed a school and several houses from which militants had been firing, ending the fighting, the officer said.
Human Rights Watch (HRW) condemned abuses by all sides in the Anbar clashes, criticising government forces for using what it alleged was indiscriminate mortar fire in civilian neighbourhoods, and militants for deploying in and attacking from populated areas.
With their seizure of Fallujah and parts of Ramadi, this is the first time militants have exercised such open control in major cities since the insurgency that followed the 2003 US-led invasion.
The al-Qaeda-linked Islamic State of Iraq and the Levant (ISIL) has been active in the Anbar fighting, but so have anti-government tribes.
At the same time, security forces have recruited their own tribal allies in the fighting that has raged in Anbar for over 10 days and killed more than 250 people.
The Iraqi Red Crescent said it had provided humanitarian assistance to more than 8,000 families across Anbar but that upwards of 13,000 had fled, while the UN said it had also managed to provide critical supplies.
Iraq was also hit by violence elsewhere today, when a suicide bomber wearing an explosives belt targeted an army recruiting centre in Baghdad, killing 23 people and wounding 30.