By Stephen Kalin

FALLUJA, Iraq (Reuters) - U.S.-backed Iraqi forces fighting Islamic State militants in Falluja are advancing toward jihadist strongholds in western districts where they expect the final push to recapture the city will take place, the Iraqi commander said on Monday.

Prime Minister Haider al-Abadi declared victory on Friday after troops reached the center of the city, an hour's drive west of Baghdad, but an official in the U.S.-led coalition said on Sunday Iraqi forces had so far taken only half of Falluja.

The operation to recapture the Iraqi city longest held by Islamic State entered its fifth week on Monday, and fighting has forced more than 85,000 residents to flee to overwhelmed government-run camps.

Iraqi forces continue to face shooting, suicide bombs and mortar attacks as they confront militants north of a road running through the city.

Heavily armored Interior Ministry police units were pressing toward Golan neighborhood, on the northwestern edge of the city, Lieutenant General Abdul Wahab al-Saidi told Reuters at his temporary command post in a southern district.

"The biggest effort now is on the western axis. If they collapse on the western axis and our forces reach Golan, you won't hear any more shots inside Falluja," he said.

Troops from Saidi's counter-terrorism force were fighting Islamic State in al-Dhubat district, further east. Fifty militants were killed there by coalition air strikes on Sunday and at least 15 others died in clashes, the commander said.

Army troops moving north from the neighborhood of Shurta had not yet entered al-Jughaifi area on the city's northern edge, while units from Baghdad operations command were advancing in the easternmost district of Askari, according to Saidi.

Sitting with other officers from the elite counter-terrorism service at a plastic picnic table littered with walkie-talkies inside an unfinished building, Saidi said the battle would end soon.

Most of the militants, including a few hundred foreign fighters, were killed or captured trying to escape with civilians, he said. Only six counter-terrorism commandos had been killed.

Government troops launched the operation on May 23 to retake Falluja, a bastion of the Sunni Muslim insurgency against U.S. forces that toppled Saddam Hussein, a Sunni, in 2003, and later against Shi'ite-led governments.

Enemies of Islamic State have launched major offensives against the jihadists on other fronts, including a push by U.S.-backed forces against the city of Manbij in northern Syria.

They amount to the most sustained pressure on Islamic State since it proclaimed a religious caliphate in 2014.

MAIN HOSPITAL CLEARED

In battle-scarred Falluja, which has witnessed more than a decade of violence -- between al Qaeda and U.S. forces and later Iraq's own security forces -- it is hard to tell how much of the visible damage has been caused by the latest fighting.

Along the route from Saidi's outpost into the city center lie the remnants of vehicle-borne suicide bombs dispatched recently by Islamic State, al Qaeda's successor. Brown scraps of metal dot the barren roadside.

The latest round of fighting, though, seems to have taken a lighter toll than previous campaigns, including the battle to retake Ramadi, the city 50 km (30 miles) further west recaptured six months ago.

Falluja has incurred fewer air strikes and Iraqi forces' quick advance to the city center last week suggested the roads were less plagued by Islamic State mines.

Saidi estimated damage to the city's infrastructure at less than five percent, which Reuters could not verify. Most houses have not yet been checked for explosives, a process that will delay the return of residents.

Government forces said on Monday the main hospital, a stronghold of militants which they surrounded a day earlier, had been partially burned but was not booby-trapped and was not sheltering suicide bombers as initially suspected.

Police checking the complex found little other than an unidentified body and the buried corpse of an Islamic State fighter.

Saidi issued a ban over his walkie-talkie against anyone re-entering the hospital, in an apparent attempt to prevent looting by undisciplined security elements.

Down a road littered with the detritus of urban warfare, a main mosque topped with azure domes, was mostly untouched. A soldier urged visiting journalists to stick to the middle of the street to avoid setting off roadside bombs.

(Writing by Stephen Kalin; Editing by Dominic Evans)