The announcements came just days after the government had declared the liberation of Fallujah, the last bastion of the Islamic State group in the sprawling western Anbar province.
With aerial support from the U.S.-led coalition, Iraqi special forces took control of the neighborhoods of al-Shurta and al-Jughaifi, special forces' Brig. Gen. Haider al-Obeidi told The Associated Press.
He said Iraqi military engineers were clearing the streets and buildings of left-over bombs.
Then, Iraqi commanders said 80 percent of the city was under their control, though clashes were still underway in its northern parts.
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In an interview with the local al-Sumaria TV late yesterday, the counterterrorism forces' chief in the Fallujah operation, Lt. General Abdul-Wahab al-Saadi, said about of 2,500 IS fighters have been killed in the offensive.
He cited Iraqi police reports as saying 1,086 IS-linked suspects have been arrested. He didn't say how many IS militants remain in Fallujah. Iraqi troops have not disclosed their losses in Fallujah, though the Islamic State group claims to have killed dozens
The operation has fueled an exodus of thousands of families, overwhelming camps for the displaced run by the government and aid groups.
UNHCR spokeswoman Ariane Rummery said that she expected that thousands more "could still be planning to leave the city."
"These escalating needs have pushed UNHCR funding into crisis levels," Rummery said. "We are exhausting available resources in Iraq to deal with the rapid developments" in Fallujah.