"I call on those heroes to move quickly to liberate the remaining part of Nineveh province, especially the right side, which is the western side of the city," Abadi said.
Mosul is divided by the Tigris River, with both sides making up about half of Iraq's second-largest city.
"The Iraqi and Peshmerga security forces fought through an elaborate defense formed over the past two years to not only keep the Iraqi security forces out, but the residents of Mosul captive," said a statement from US Central Command Combined Joint Task Force-Operation Inherent Resolve.
The US statement lauded Iraqi efforts to keep civilians safe as it fended off mortar, sniper and drone attacks, while ISIS used human -- including child shields -- and stored weapons in hospitals, mosques and schools.
"To see how far the Iraqis have come since 2014, not only militarily, but in their ability to put their differences aside and focus on a common enemy, gives real hope to the people of Iraq that after years of fighting and instability, peace and security are attainable," said Lt. Gen. Stephen Townsend, the task force's commanding general.
Experts said last week the terrorist organization had been confined to Mosul's urban centers and was on the cusp of losing the city's east bank. Still, the battle to wrest control of the west side could drag on into March.
Losing its last major Iraqi stronghold would be a huge blow to ISIS, which, according to US Air Force Col. John L. Dorrian, has already lost Ramadi, Falluja, Hit, Qayyara and Sharqat. Tuesday's statement also mentions Kirkuk and Tikrit.
Earlier this month, Iraqi-led forces made significant gains
, seizing the Mosul's university, a government complex and parts of Tigris' east bank, said Lt.
Gen. Abdul Wahab al-Saadi of the Iraqi counterterrorism forces.
ISIS seized control of Mosul in 2014. Ousting the group would be a challenge for any of the world's militaries, Townsend said.
The new government complex in eastern Mosul as well as several Mosul University buildings at one time had been a headquarters and killing ground for ISIS fighters.
Iraqi forces discovered large amounts of suspected chemical substances and a "bomb factory" at the university, said Sabah al Numan, spokesman for Iraqi counterterrorism forces.
ISIS militants have used chemical weapons such as mustard gas
, according to Iraqi and US officials.
About 10 days ago, counterterrorism forces reached the east bank of the Tigris for the first time since the large-scale military operation to reclaim Mosul began in October.
They did so in spite of ISIS destroying all five of Mosul's bridges crossing the Tigris
to slow the advance of Iraqi troops toward the city's western side, said Lt. Gen. Abdul Amir Rasheed Yarallah, commander of Iraqi forces in Nineveh province.
The bridges had already been damaged in airstrikes carried out by the US-led coalition. In previous instances where ISIS destroyed bridges, Iraqi forces built temporary or floating structures, slowing down troops but not deterring their advance.
"There is still a long way to go before ISIL is completely eliminated from Iraq, and the fight for Western Mosul is likely to be even tougher than the Eastern side," Townsend said in the Tuesday statement, using another name for ISIS. "But the (Iraqi security forces) have proven they are both a professional and formidable fighting force, and I have every confidence that ISIL's days are numbered in Iraq."
On behalf of the coalition, Townsend congratulated the Iraqi troops, wishing them "good luck and Allah's blessings for the fight on the west side that lies ahead."
Leader in cross hairs?
News of advances made by Iraqi forces comes days after outgoing US Defense Secretary Ash Carter said during a PBS interview that ISIS leader Abu Bakr al-Baghdadi's "days are numbered."
"I'm just confident. I don't want to say any more than that, but I wouldn't want to be a senior ISIL leader," Carter said. "Many of them have died already. The more we do, the more we learn about where they are. So his days are numbered, and that's true of all the rest of the leadership."
An American official said December 29, "In the last few weeks we've been aware of some of Baghdadi's movements."
The official had access to some of the most recent US government reporting on ISIS at the time, but because of the sensitivity of the information, the official declined to provide more details. That includes whether intelligence indicates if Baghdadi is in Syria or Iraq.
According to Iraq's Shiite-led Popular Mobilization Units, or PMUs, Baghdadi is hiding in an area west of Mosul.
"We've received information that Abu Bakr al-Baghdadi is hiding between the towns of al Baaj and Al-Qairawan, near the Iraqi-Syrian border," PMU spokesman Ahmed al Assadi has said.
Last month, American officials increased the reward for information leading to Baghdadi's capture to $25 million from $10 million.