A United Nations envoy has warned of a dire humanitarian situation in western Iraq, as Washington piled pressure on the Iraqi premier to quell an Al-Qaeda surge in the violence-plagued country.

More than 13,000 families have fled the militant-held city of Fallujah, the Iraqi Red Crescent said Wednesday, as masked gunmen held the city which is locked in a days-long standoff with the army.

And though traffic police returned to its streets, some shops reopened and more cars could be seen, Fallujah was still rocked by clashes and shelling, after an Al-Qaeda-linked group urged Sunnis to keep fighting the Shiite-led government.

Fallujah and parts of nearby Ramadi, both in the western province of Anbar, have been outside government hands for days -- the first time militants have exercised such open control in major cities since the insurgency that followed the 2003 US-led invasion.

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 and were living with relatives, or in schools or other public buildings.

"There is a critical humanitarian situation in Anbar province which is likely to worsen as operations continue," Nickolay Mladenov, the UN special envoy to Iraq, said in a statement.

"The situation in Fallujah is particularly concerning as existing stocks of food, water and life-saving medicines begin to run out."

US Vice President Joe Biden called Nuri al-Maliki for the second time this week, mounting pressure on the Iraqi premier over the Al-Qaeda surge.

Biden urged Maliki to "continue the Iraqi government's outreach to local, tribal, and national leaders," following the loss of Fallujah to Islamist insurgents, the White House said in a statement.

Spokesman Jay Carney said Washington was pressing Maliki, a Shiite, to focus on political reconciliation as well as take military action to expel Al-Qaeda-inspired groups from Fallujah and Ramadi, both Sunni bastions in Anbar province, once liberated from extremists by US troops.

US Secretary of State John Kerry pressed home a similar message in a call with Iraqi Foreign Minister Hoshyar Zebari, the State Department said.

The calls came amid discussion in Washington foreign policy circles about who should take the blame for the resurgence of Al-Qaeda.

Critics of the White House blame President Barack Obama for failing to agree a deal with Maliki's government to leave a residual US force behind after withdrawing all American troops from Iraq at the end of 2011.

On the ground a suicide bomber detonated an explosives-rigged vehicle in Ramadi on Wednesday, killing two Sahwa anti-Al-Qaeda militiamen and wounding four others, while another bombing damaged a bridge in the city.

And two areas of Fallujah saw brief clashes and shelling, witnesses said, but it was not immediately clear who was involved in the fighting.

More than 250 killed

The Al-Qaeda-linked Islamic State of Iraq and the Levant (ISIL) has been active in Fallujah, but so have anti-government tribes.

The security forces have meanwhile recruited their own tribal allies in the fighting that has raged in Anbar for more than a week and killed more than 250 people.

Near the provincial capital Ramadi, soldiers backed by helicopters battled gunmen in the Khaldiyah area, a police captain said.

On Tuesday, ISIL spokesman Abu Mohammed al-Adnani urged Iraqi Sunnis to continue battling government forces.

"Do not lay the weapon down, because if you put it down this time, the (Shiites) will enslave you and you will not rise again," Adnani said in a recording.

A military spokesman has said an assault on Fallujah was on hold for fear of civilian casualties.

Attacking the Sunni-majority city would be a significant test for Iraqi security forces, who have yet to undertake such a major operation without the backing of US troops, who withdrew in December 2011.

It would also be extremely sensitive politically, as it would inflame already high tensions between the Sunni Arab minority and the government.

Both Ramadi and Fallujah were insurgent strongholds in the years after 2003, and Fallujah was the target of two major assaults in which US forces saw some of their heaviest fighting since the Vietnam War.

They eventually wrested back control of Anbar with the support of Sunni tribesmen who formed the Sahwa (Awakening) militias, which allied with US troops against Al-Qaeda from late 2006.

But Sunni militants have regained strength, bolstered by the war in neighbouring Syria and widespread Sunni Arab anger with the federal government.

Fighting erupted near Ramadi on December 30, when security forces cleared a year-old Sunni protest camp.

The violence spread to Fallujah, and militants moved in and seized the city and parts of Ramadi after security forces withdrew.

Unrest elsewhere in Iraq on Wednesday killed 22 people, including 12 soldiers and six police, officials said.