The militants have already been driven out of all of the towns they once held but Prime Minister Haider Al-Abadi has said he will not proclaim victory until they have been cleared from the western desert bordering Syria.
The Hashed Al-Shaabi (Popular Mobilization) paramilitary force said its fighters had taken control of 77 villages and hamlets since the launch of the offensive on Thursday morning.
It said five militants had been killed south of the ancient desert city of Hatra but otherwise Daesh had put up little resistance.
The Hashed said that its fighters overran an airfield in the same area, where they discovered underground warehouses used by the militants.
Air support for the offensive, which also involves the army and federal police, has so far been provided exclusively by the Iraqi air force.
The US-led coalition, which has provided air support for other offensives against Daesh in Iraq, said it carried out no strikes on Thursday.
At its peak in 2014, Daesh ruled over 7 million people in a territory as large as Italy encompassing large parts of Syria and nearly a third of Iraq.
It is now being flushed out of its last desert hideouts in Iraq and under attack by Russian-backed government forces and US-backed Kurdish-led fighters in its last pockets of control in Syria.