Iraqi forces seized a damaged Mosul bridge on Monday which could link up their units on either side of the Tigris river, as thousands of civilians fled the fighting for Isis's remaining stronghold in the west of the city.

US-backed army and police units advanced through populated western districts, fighting tough street battles, and announced they had captured Mosul's southernmost bridge.

Once repaired, the bridge could help bring reinforcements and supplies from the eastern side, piling pressure on the militants dug in the western side among 750,000 civilians.

Iraqi forces captured eastern Mosul in January, after three months of fighting. They launched their attack on the districts that lie west of the Tigris a week ago.

If they defeat Isis in Mosul, that would crush the Iraq wing of the caliphate that the group's leader Abu Bakr al-Baghdadi declared in 2014 over parts of Iraq and neighbouring Syria. The US commander in Iraq has said he believes US-backed forces will recapture both Mosul and Raqqa – Isis's Syria stronghold – within six months.

Since government forces broke through the city's southern limits last Thursday, more than 10,000 civilians have fled Isis-held areas, seeking medical assistance, food and water, Iraqi commanders said.

About 1,000 civilians arrived in the early hours of Monday at the sector held by the Counter Terrorism Service (CTS), the wounded taken to the clinic of this elite unit, while men were screened to make sure they are not Isis members.

Among the people treated at the CTS clinic was a little girl with a blood on her face and a woman with shrapnel embedded in her hand, lying immobile, apparently unconscious.

An old man who came with them said about 20 people were sheltering in their house when it was hit by an air strike two days ago in the south-western Maamoun district.

Those who managed to escape have had to walk through the desert for at least an hour to reach government lines.

Several thousand militants, including many who travelled from Western countries to join up, are believed to be still in Mosul, prepared for a fierce standoff amid a remaining civilian population of 750,000.

The United Nations World Food Programme said on Monday it was extremely concerned about dire humanitarian situation facing families in western Mosul.

A reporter saw several trucks teeming with people, lifting columns of sand and dust as they drove away from the city. One had two women and infants riding in the cabin. The rest stood on the open bed, held on to the truck from outside, or sat on top of the cabin. "They booby trapped our homes and our cars," said an old woman.

A Western volunteer medic at the CTS clinic said a boy with a gunshot wound that shattered his knee was among those treated on Monday, and a pregnant woman who had both legs amputated.

"Most of those who arrive to this point are hungry and thirsty and suffering neglect, and need medical care," CTS Brigadier General Salman Hashim said.

Army, police, CTS and Rapid Response units forces attacking Isis in west Mosul are backed by air and ground support from US-led coalition, including artillery fire. US personnel are operating close to the front lines to direct air strikes.

Iraqi troops have already captured the southern and western accesses to western Mosul, dislodging the militants from the airport, a military base, a power station and three residential districts, al-Maamoun, al-Tayyaran and al-Josaq, according to military statements.

"The more we advance, the fiercer the resistance," said Lt Colonel Abdel Amir al-Mohammadawi, from the rapid response units that are fighting near the southernmost bridge, one of five spanning the Tigris.

All of them were damaged in strikes by the US-led air coalition, and later by Isis fighters trying to seal off the western bank still under their control.

Iraqi forces have reached less than than a mile from the old city centre and the main government buildings, the capture of which would effectively mean the fall of Mosul.

The militants are using mortar, sniper fire, booby traps and suicide car bombs to fight off the offensive.

They are facing a 100,000-strong force made up of Iraqi armed forces, regional Kurdish Peshmerga fighters and Iranian-trained Shia Muslim paramilitary groups.