By John Davison

MOSUL (Reuters) - Iraqi forces battling Islamic State faced tough resistance from snipers and mortar rounds on Monday as they tried to advance on Mosul's Old City and a bridge across the Tigris river in their campaign to retake the western part of the city.

Progress by Rapid Response units was slowed by heavy rain on Monday morning, but they were only 100 meters (yards) from the Iron Bridge which connects the Old City with the eastern side of Mosul, military officials said.

Since starting the campaign in October, Iraqi forces with U.S.-led coalition support have recaptured eastern Mosul and around 30 percent of the west from militants who are outnumbered but fiercely defending their last stronghold in Iraq.

Federal police and Rapid Response brigades, an elite Interior Ministry unit, said at the weekend they had entered the Bab al-Tob area of the Old City, where fighting is expected to be toughest because of its narrow alleyways where armored vehicles cannot pass. But the advance there stalled on Monday.

"Due to the bad rainy weather, operations have been halted for now. We are facing stiff resistance from the Daesh (Islamic State) fighters with sniper shots and mortars," a rapid response unit officer told Reuters.

A Reuters reporter on the ground said mortar rounds and sniper fire hit near the Mosul museum area, which Iraqi forces had seized from militants a few days ago.

Troops exchanged fire with snipers, while trying to drag blinds made of blankets and curtains across streets to obscure their movements. Heavy explosions later hit a hotel where Islamic State gunmen had been returning fire.

"We moving on the old bridge...and then we will free that area and hopefully in a few days we'll liberate the west side of Mosul," said one Iraqi captain fighting there.

Federal police forces also fighting in areas close to the Iron Bridge were battling pockets of militants in Bab al-Tob district and carrying out house to house searches, a federal police commander's spokesman said.

A military statement said on Monday that Iraqi elite Counter Terrorism Service troops, known as CTS, managed to retake al-Nafut district of west Mosul.

TENS OF THOUSANDS FLEEING

As many as 600,000 civilians are caught with the militants inside Mosul, which Iraqi forces effectively sealed off from the remaining territory that Islamic State controls in Iraq and Syria. The Iraqi forces include army, special forces, Kurdish peshmerga and Shi'ite militias.

More than 200,000 Mosul residents have been displaced since the start of the campaign in October, of which more than 65,000 fled their homes in the past two weeks alone, according to the International Organization for Migration.

Residents fled from the Wadi Hajjar district carrying bags and belongings through the muddy streets to areas controlled by the army before being transported out by truck.

"It was a nightmare and we are finally done with it. It cannot be described. There is no water and there is no food and nothing," said Ghanem Mohamed, one of the residents who fled.

Capturing the Iron Bridge would mean Iraqi forces control three of the five bridges spanning the Tigris river between eastern and western Mosul, all which have been damaged by Islamic State fighters and U.S.-led air strikes. The southernmost two have already been retaken by Iraqi forces.

Losing Mosul would be a major strike against Islamic State. It is by far the largest city the militants have held since their leader Abu Bakr al-Baghdadi proclaimed himself leader of a self-styled caliphate spanning Iraq and Syria from a mosque in Mosul in the summer of 2014.

As Iraqi forces slowly take territory from the group, more evidence is emerging of war crimes committed by the Sunni Islamist militants, who targeted Shi'ite Muslims and religious minorities as well as opponents from their own sect.

(Additional reporting by Ahmed Rasheed in Baghdad; writing by Patrick Markey; Editing by Dominic Evans)