* US-led coalition says Iraqis don't have full control yet
* More than 53,000 people displaced since assault began in May
* Aid groups: Military surge overwhelming camps with refugees
Iraqi forces recaptured the municipal building in Fallujah from Islamic State militants, the military said on Friday, nearly four weeks after the start of a US-backed offensive to retake the city an hour's drive west of Baghdad.
The ultra-hardline militants still control a significant portion of Fallujah, where the conflict has forced the evacuation of most residents and many streets and houses remain mined with explosives.
A spokesman for the US-led coalition backing Baghdad's quest to recover large swathes of western and northern Iraq from Islamic State told Reuters that government forces were "close (to the building) but don't have control yet".
A military statement said the federal police had raised the Iraqi state flag above the government building and were continuing to pursue insurgents.
A Reuters photographer in a southern district of Fallujah said clashes involving aerial bombardment, artillery and machine gun fire were continuing. Clouds of smoke could be seen rising up from areas closer to the city centre.
Heavily armed Interior Ministry police units were advancing along Baghdad Street, the main east-west road running through the city, and commandos from the counter-terrorism service (CTS) had surrounded Fallujah hospital, the statement said.
Sabah al-Numani, a CTS spokesman, said on state television that snipers holed up inside the hospital, considered a nest of militants, were resisting but the facility was expected to be retaken within hours.
Government forces, with air support from the US-led coalition, launched a major operation on May 23 to retake Fallujah, an historic bastion of the insurgency against US forces that toppled Saddam Hussein in 2003 and the governments that followed.
The city is seen as a launchpad for recent Islamic State bombings in the capital, making the offensive a crucial part of the government's campaign to improve security.
US allies would prefer to concentrate on Islamic State-held Mosul, Iraq's second largest city that is located in the far north of the country.
Enemies of Islamic State have uncorked major offensives against the jihadists on other fronts, including a thrust by US-backed forces against the city of Manbij in northern Syria.
The offensives amount to the most sustained pressure on IS since it proclaimed a caliphate in 2014.
Islamic State has begun allowing thousands of civilians trapped in central Fallujah to escape and the sudden exodus has overwhelmed displacement camps already filled beyond capacity.
More than 6,000 families left on Thursday alone, according to Fallujah Mayor Issa al-Issawi, who fled the IS seizure of Fallujah two years ago. He told Reuters on Friday: "We don't know how to deal with this large number of civilians."
The number of displaced people as of Thursday surpassed 68,000, according to the United Nations, which recently estimated Fallujah's total population at 90,000, only about a third of the total in 2010.
Witnesses said Islamic State had announced via loudspeakers that residents could leave if they wanted, but it was unclear why the group changed tact after clamping down on civilian movement only a few days ago.
The Norwegian Refugee Council (NRC), which has been providing aid to displaced people, said escapees reported a sudden retreat of IS fighters at key checkpoints inside Fallujah that had allowed civilians to leave.
Humanitarian needs were expected to increase dramatically in the coming hours, swamping the resources of foreign aid groups and the government as they struggle with funding shortfalls.
"Aid services in the camps were already overstretched and this development will push us all to the limit," said NRC country director Nasr Muflahi.
Islamic State, which by US estimates has been ousted from almost half of the territory it seized when Iraqi forces partially collapsed in 2014, has used residents as human shields to slow the military's advance and help avoid air strikes.
Defence Ministry spokesman Naseer Nuri said the surge in displaced people was "proof that (Islamic State) has lost control over the city and its residents".
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