The Iraqi army says it hopes to capture the last ISIL enclaves outside Mosul's Old City in the next 72 hours, as fighting continued on the second day of a new push to drive the armed group from its remaining pockets in its stronghold in Iraq.

Security forces and their allies on Saturday ramped up an operation that began seven months ago, launching an assault from three fronts to close in on ISIL positions in the neighbourhoods of Shifa, Zinjili and Saha.

The head of the Iraqi operation said his forces are pushing ISIL fighters back, despite facing strong resistance.

"The fighting is fierce, but our forces managed to kill a number of ISIL fighters, including four suicide attackers," Major General Maan al-Saadi, said.

"The battle and the advances are continuing. I guess in 72 hours we will be able to take full control of these neighborhoods, and then we can say that we have besieged the whole Old City of Mosul."

At least 15 Iraqi forces soldiers were killed in the first day of the offensive, security sources said, adding that the armed group had deployed snipers, suicide car bombers and suicide attackers on foot.

More than 30 ISIL fighters were also killed in heavy clashes.

Speaking on condition of anonymity to the Associated Press news agency, two Iraqi military officers described the advance on Mosul's Old City as "cautious" and the clashes on Sunday as "sporadic."

READ MORE: Iraqi artist depicts life under ISIL

The three neighbourhoods targeted in the latest offensive, located north of the Old City, are seen as the last areas that separate Iraqi forces from ISIL's last bastion.

They have narrow streets and closely-spaced buildings, posing significant challenge to the Iraqi forces seeking to oust ISIL, also known as ISIS.

Military and political analysts expect that it could take several days for the Iraqi forces to gain hold of these territories, before making their final push for the Old City.

Separately, the Popular Mobilization Forces - Shia militia fighters that have official status from Baghdad -entered the fourth day of an operation aimed at clearing the main highway that links the recently retaken district of Qairawan to al-Ba'aj, west of Mosul.

The fighters are are planning to establish a secure strategic military line for their forces before they reach the Iraqi-Syrian borders in order to cut off ISIL supply lines and infiltration.

The start of the new push towards the Old City coincided with the start of the holy fasting month of Ramadan.

Residents in the Old City sounded desperate in telephone interviews over the past few days.

"We're waiting for death at any moment, either by bombing or starving," one told the Reuters news agency, asking not to be identified. "Adults eat one meal a day, either flour or lentil soup."

As the fighting intensified, the United Nations expressed "deep concern" for the hundreds of thousands of civilians trapped behind ISIL lines.

"Although the UN is not present in the areas where fighting is occurring, we have received very disturbing reports of families being shut inside booby-trapped homes and of children being deliberately targeted by snipers," Stephen O'Brien, the UN's under-secretary general for humanitarian affairs, said in a statement on Saturday.

The UN last week said up to 200,000 more people - half of them children - could flee Mosul as fighting moves to the Old City.

About 700,000 people, or a third of the pre-war city's population, have already fled, seeking refuge either with friends and relatives or in camps or neighbouring towns and cities.

In Erbil, east of Mosul, thousands of people have been brought to receive treatment, including many children.

"I was trying to escape from the Old City at 3am and two bullets hit me," a young boy told Al Jazeera from an Erbil hospital. "One ripped through my leg, the other in my groin."

Al Jazeera's Osama bin Javaid, reporting from Erbil, said the child's story is repeated hundreds of times in each hospital of the city.

"Almost all children in this ward are from the embattled city of Mosul. Most of them have been wounded as they try to reach towards safety, but in densely populated Mosul, where bombs continue to fall and snipers continue to shoot, there is no place that is safe," he said.

READ MORE - In west Mosul, 'nowhere is safe for civilians'

ISIL overran large areas north and west of Baghdad in 2014, but Iraqi forces backed by US-led air raids have since regained much of the territory they lost to the group.

Iraqi forces launched the major operation to retake Mosul in October last year, fighting their way to the city and retaking its eastern side before setting their sights on its smaller but more densely populated west.

The battle has taken a heavy toll on civilians, pushing hundreds of thousands to flee, while hundreds more have been killed or wounded.