Iraqi forces have broken into jihadist-held Mosul and recaptured neighbourhoods inside the city, but a month into their offensive, there are still weeks or more of potentially heavy fighting ahead.

Iraq launched the massive operation to recapture Mosul, which has been held by the Islamic State group since 2014, on October 17, advancing on the city from the north, east and south.

Iraq's elite Counter-Terrorism Service and regular army soldiers have retaken areas on its eastern side from IS, but forces on other fronts have yet to enter the city.

"Fighting inside of the city is likely to grow increasingly more difficult as the (Iraqi forces) progress through Mosul's more modern eastern neighbourhoods towards its older, denser centre," said Patrick Martin, an Iraq analyst at the Institute for the Study of War.

The jihadists' "use of tunnels, human shields, suicide car bombs, and other attack capabilities will become harder to counter as fighters are forced into closer quarters by the nature of the city's geography," Martin said.

But Michael Knights of the Washington Institute for Near East Policy cautioned that the nature of IS defences deeper in the city remains unclear.

"We don't know if the current fighting represents the most heavily fortified (and) defended neighbourhoods or a precursor to the toughest fighting" in the old city, Knights said.

"We could be fighting through a hard crust with a soft inside, or it could be all hard," he said.

- Third of eastern Mosul retaken -

The United States has said that hundreds of IS fighters have been killed since the offensive began, but those figures cannot be independently confirmed.

Iraqi forces have not released information about their casualties.

Interior ministry spokesman Saad Maan has said that more than a third of eastern Mosul has been retaken, but that would still leave a significant part of the east and all of the west -- which is expected to put up the toughest resistance -- in IS hands.

Iraqi Kurdish forces played a role in the early days of the Mosul operation, but said after recapturing the town of Bashiqa to the city's east that their offensive participation in the battle is over.

Interior ministry forces, including the elite Rapid Response Division, as well as soldiers on the southern front have had the longest way to go to reach Mosul, but they are now nearing the airport on its southern outskirts.

A US-led coalition is carrying out air and artillery strikes in support of Iraqi forces, and said it has targeted the jihadists with thousands of bombs, missiles, rockets and shells.

The Popular Mobilisation forces, an umbrella group for pro-government paramilitaries that is dominated by Iran-backed Shiite militias, began an operation to retake the town of Tal Afar, on the western approach to Mosul, at the end of last month.

They have since retaken a series of villages as they push west toward Tal Afar, which was seized by IS along with swathes of other territory north and west of Baghdad in 2014.

Inside Mosul, Iraqi forces "still must advance towards the Tigris River that bisects Mosul as well as clear the entire western half of the city," Martin said.

Knights said the level of IS resistance in Mosul "is much higher" than that faced in the battles for Tikrit and Fallujah, two cities that Iraqi forces recaptured from the jihadists.

- Danger for civilians -

Civilians are already in danger of being caught in the crossfire and used as human shields by IS, a risk that may grow as Iraqi forces push deeper into Mosul.

Mosul residents told AFP that IS had sought to forcibly gather civilians for apparent use as human shields, while the UN rights office said it had received reports of the jihadists seizing thousands of people and executing scores.

Aid workers have warned that up to a million people could be displaced by the Mosul operation.

While those numbers have yet to materialise, more than 56,000 people have been displaced since the operation began, according to the International Organization for Migration (IOM).

Some people have remained in Mosul after areas have been recaptured from IS, but civilians, especially children who go out to play, are still at risk even after the jihadists are gone from the immediate area.

One boy was killed and another had his leg ripped open by mortar fire in east Mosul earlier this week, while a field clinic in the area also recently treated a young girl whose leg was hit by shrapnel.

"Families from Mosul are in perilous situations, both those who are still in their homes and those fleeing amidst heavy fighting," said IOM Iraq chief of mission Thomas Lothar Weiss.

"They have witnessed extreme violence and continue to be at risk."