By Saif Hameed and Stephanie Nebehay

BAGHDAD/GENEVA (Reuters) - U.S. forces backing an Iraqi army campaign against Islamic State in Mosul carried out an air strike on a bridge spanning the Tigris river, restricting militant movements between western and eastern parts of the city, a U.S. official said on Tuesday.

U.S.-trained Iraqi Counter-Terrorism Service forces are pushing deeper into east Mosul, the last major city controlled by the Sunni hard-line group in Iraq, while army and police units, Shi'ite militias and Kurdish fighters surround it to the west, south and north.

Militants have steadily retreated into Mosul from outlying areas. The army's early advances have slowed as militants dig in, using the more than 1 million civilians inside the city as a shield, moving through tunnels, and hitting troops with suicide bombers, snipers and mortar fire.

Five bridges span the Tigris that runs through Mosul. They have all been mined and boobytrapped by militants who took over the city two years ago as they swept through northern Iraq and declared a caliphate in parts of Iraq and neighboring Syria.

Despite planting the mines, Islamic State fighters have so far been able to continue using those bridges which have not yet been destroyed by air strikes.

Air Force Colonel John Dorrian, a Baghdad-based spokesman for the U.S.-led coalition, said on Tuesday an air strike hit the number four bridge, the southernmost, in the past 48 hours.

"This effort impedes Daesh's freedom of movement in Mosul. It inhibits their ability to resupply or reinforce their fighters throughout the city," he said using an Arabic acronym for the militant group.

A month ago, a U.S. air strike destroyed the No. 2 bridge in the center of the city and two weeks later another strike took out the No. 5 bridge to the north.

The United Nations' International Organisation for Migration expressed concern that the destruction of the bridges could obstruct the evacuation of civilians.

"That is a concern of IOM because this is going to leave hundreds of thousands without a quick way out of the combat," spokesman Joel Millman told reporters in Geneva.

COMMANDERS CAPTURED

The battle for Mosul, launched five weeks ago, is turning into the largest military campaign in more than a decade of conflict in Iraq since the 2003 U.S.-led invasion that toppled Saddam Hussein.

The Iraqi military estimates around 5,000 Islamic State fighters are in Mosul. A 100,000-strong coalition of Iraqi government forces, Kurdish fighters and Shi'ite paramilitary units is surrounding the city.

Mosul's capture would be a major step towards dismantling the caliphate, and Islamic State leader Abu Bakr al-Baghdadi, believed to have withdrawn to a remote area near the Syrian border, has told his fighters to stay and fight to the end.

Counter terrorism units and an army armored division are the only forces to have breached the city limits from the eastern side. Other army and federal police units have yet to enter the northern and the southern sides.

A Kurdish security source said on Tuesday four Islamic State commanders were captured in a U.S. special operation near Baaj, a town close to the Syrian border. Baghdadi was not among them. The coalition did not confirm the operation.

Islamic State said it launched an attack on the north-western front of Mosul, seizing a duty free zone and oil depots located a dozen kilometers from the city limits. The army did not confirm the claim.

Iranian-backed militias have captured the Tal Afar air base, west of Mosul, part of their campaign to choke off the route between the Syrian and Iraqi parts of the caliphate Islamic State declared in 2014.

The number of people displaced by the fighting in and around Mosul has slightly decreased, an indication that some people have began returning home in places retaken by government forces, according to the IOM.

"68,112 displaced is actually a downtick from couple of days ago," said Millman. It's "worth noting because it indicates that some people are already starting to return to safe areas in the region."

The number of registered displaced people was over 68,500 on Monday. The figure does not include the thousands of people rounded up in villages around Mosul and forced to accompany Islamic State fighters to cover their retreat towards the city. 

An Iranian-backed Shi'ite group taking part in the offensive denied a Human Rights Watch report that it detained and beat 10 shepherds, including a boy, from a village near Mosul on November 3.

Human Rights Watch on Tuesday said the League of the Righteous, or Asa'ib Ahl al-Haq, also stole the village's entire flock of sheep.

"These reports are completely wrong, they aim to stall the operation" against Islamic State, said Asa'ib's military spokesman Jawad al-Talabawi.

The Iraqi government has not published an overall death toll for the offensive, whether among military or civilians. The warring sides claim to have killed thousands in enemy ranks.

(Writing by Patrick Markey and Maher Chmaytelli; Editing by Dominic Evans)