Iraqi special forces battled the Islamic State group on Friday in west Mosul, where a commander said jihadist resistance is showing signs of weakening under repeated assaults.
The jihadists are also facing simultaneous offensives in Syria by government forces, Turkish-backed rebels and a US-supported alliance of Kurdish and Arab fighters, piling more pressure on IS.
But the battle for Mosul's Old City -- which could see some of the toughest fighting of the operation -- has not yet begun, nor has fighting inside the city of Raqa, IS's main bastion in Syria.
Iraq's Counter-Terrorism Service assaulted the Al-Amil al-Oula neighbourhood of west Mosul early on Friday, and were battling the jihadists inside it, said Staff Major General Maan al-Saadi, a CTS commander.
"The enemy fought fiercely in the first defensive line from Al-Ghazlani base to Wadi Hajar and to Al-Sumood neighbourhood,"Saadi said, referring to areas that Iraqi forces have recaptured since launching their assault on west Mosul on February 19.
But"after these neighbourhoods, after we broke the defensive line, they lost many fighters,"he said.
"The enemy has begun to collapse. They have lost many of their combat capabilities. Today, the enemy sent (car bombs), but not in the numbers that they sent at the beginning of the battle."
In another sign that the jihadists are feeling the squeeze, their chief Abu Bakr al Baghdadi was reported to have abandoned Mosul, leaving local commanders behind to oversee IS's defence of the city.
"He was in Mosul at some point before the offensive.... He left before we isolated Mosul and Tal Afar,"a town to the west, a US defence official said.
"He probably gave broad strategic guidance and has left it to battlefield commanders."
Baghdadi, who declared IS's cross-border"caliphate"at a Mosul mosque in 2014, urged supporters in an audio message in November to make a stand in the city rather than"retreating in shame".
Iraqi forces launched their operation to retake Mosul in October, and recaptured the whole east bank of the Tigris River that runs through it in January.
They then set their sights on the smaller but more densely populated west side of the city. More than 215,000 people are currently displaced as a result of the battle for Mosul, according to the International Organization for Migration. Others fled their homes but later returned.