Iraqi forces yesterday battled the last remaining Islamic State group fighters in Mosul's Old City, where the UN said up to 20,000 civilians are trapped and in "extreme danger".

In neighbouring Syria, US-backed forces are facing fierce resistance from IS as they wage a parallel offensive to recapture the city of Raqa from the jihadists.

The fighting in both cities is taking place in densely populated residential neighbourhoods, which are sustaining a high civilian death toll and heavy damage as diehard jihadists put up a desperate last stand.

"Our estimate at this stage is that in the final pockets of (Mosul's) Old City, there could be as many as 15,000 civilians, possibly even as high as 20,000," UN humanitarian coordinator in Iraq Lise Grande told AFP.

"They're in extreme danger from bombardment, from artillery crossfire. The (IS) fighters that are still there are still directly targeting civilians if they try and leave," she said.

More than eight months after the start of the operation to retake Mosul from IS, the jihadists have gone from fully controlling the city to holding a small pocket on the west bank of the River Tigris that flows through it.

On Tuesday, Prime Minister Haider al-Abadi congratulated the security forces on the "achievement of this major victory in Mosul".

But troops and police are facing a rising number of suicide attacks, including some by female bombers, as the campaign nears its climax.

The battle for Mosul has pushed 915,000 to flee their homes, nearly 700,000 of whom are still displaced.

"We exceeded our worst case scenario more than a month ago. In our very worst-case scenario, we thought that 750,000 people would flee," Grande said.

The damage caused by the fighting in west Mosul -- and the cost of addressing it -- is huge.