Mosul had been the last major IS stronghold in Iraq, but recent offenses have cornered the jihadists into select pockets of the old city.
"Our units are continuing to advance. and entered Al-Saha al-Oula and Al-Zinjili and Al-Shifaa neighborhoods and the Republican Hospital", said Joint Operations Command spokesman Yahya Rasool.
They have narrow streets and closely-spaced buildings which pose significant challenge to the forces seeking to oust ISIL, also known as ISIS.
Little has changed at the scene of the al-Jadida airstrike that killed more than 100 civilians more than two months ago and ratcheted up pressure on the USA -led coalition to address allegations of mounting civilian casualties in the fight against the Islamic State group. That allowed the Iraqi army to rack up many quick gains in the fight against the militant group.
Iraqi officials are demanding that the U.S. compensate the victims of the March 17 airstrikes in Mosul, following the Pentagon finally admitting that they not only attacked the buildings, but had killed at least 105 civilians in the process. They spoke on condition of anonymity in line with regulations. Civilians too have been caught in the crossfire, leaving hundreds of thousands displaced and thousands dead.
Iraqi forces, backed by global coalition, have been fighting to drive out IS militant from the western side of Mosul, but several neighborhoods, including the densely-populated old city center, are still under control of the extremist militant.
After almost eight months of fighting Iraqi forces appear poised to retake what was once the Islamic State's largest stronghold in Iraq and the birthplace of its caliphate.
An average of 10,000 people were reported to flee conflict areas on a daily basis in Western Mosul with 16, 000 fleeing their homes last Thursday, the largest movement of civilians in a single day, the International Organization for Migration (IOM) said this week.
Within hours of the offensive's start, the government announced two military officers had been killed in Shafaa.
The Pentagon denies this and says its rules of engagement remain unchanged and insists its precision-targeting abilities are the best in the world.
The death cult now controls only the Zanjili and Shafaa districts and the labyrinthine Old City - the site of the symbolic Al-Nouri mosque where Isis leader Abu Bakr Al-Baghdadi first declared his caliphate in 2014.
Earlier this week, the military said it had dropped "hundreds of thousands of leaflets" on IS-held areas of Mosul urging "citizens to exit via safe corridors towards security forces".
The camp adjacent to the site is at full capacity, and many families are being moved to other camps, with large numbers opting to go to areas east of Mosul where they stay with families, friends or in rental accommodations, says IOM.
The insurgents are also retreating in Syria, mainly in the face US-backed Kurdish-led forces. "Majority 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.