On Tuesday, Iraqi forces took another Kurdish-held town -- Sinjar, near the Iraq-Syria border -- that the Kurds wrested from ISIS control in 2015, said Brig. Gen. Yehya Rasool, spokesman of the Iraqi Joint Operation Command. Rasool promised a detailed statement, outlining all the locales retaken by Iraqi forces, later Tuesday.
Iraqi Prime Minister Haider al-Abadi made clear his forces would eventually come back for Kirkuk, and he ordered the operation to "secure" it Sunday, weeks after the Kurds held an independence referendum claiming the disputed city as their own.
An estimated 61,200 people were displaced around Kirkuk the past two days, according to the United Nations Office for the Coordination of Humanitarian Affairs.
Emergency food rations and other aid is being provided to families sheltering in public spaces, the office said in a statement Tuesday.
"Authorities have confirmed that the majority of displaced families from Kirkuk are already returning to their homes in Kirkuk city," said Lise Grande, UN humanitarian coordinator for Iraq. "Most of the families who are still displaced are staying with families and host communities."
"We call on all parties to do everything possible to shield and protect all civilians impacted by the current situation," she said.
At least 16 Kurdish fighters were killed in the operation, Kurdish Peshmerga commanders said, saying Iraqi forces used US-supplied weapons against them.
Massoud Barzani, president of the Kurdish Regional Government in Iraq, issued a statement saying that the events in Kirkuk were the work of "a few people in Kurdish politics and the government of Baghdad. As a result, Peshmerga were ordered to pull back."
A dilemma for US
President Donald Trump insisted Washington would not take sides in the dispute.
"We don't like the fact that they're clashing. We're not taking sides," Trump told reporters at the White House. "We've had for many years a very good relationship with the Kurds as you know, and we've also been on the side of Iraq, even though we should have never been there in the first place."
Asked who controls Kirkuk, a spokesman for the US-led coalition, Col. Ryan Dillon, said that was a question for the Iraqi government and the coalition's focus is fighting ISIS.
"We don't have advisers and people on the ground with the Iraqi security force or the Peshmerga elements that are in and around Kirkuk," Dillon said.
Meanwhile, Germany was in talks with the coalition after temporarily suspending its efforts to train the Peshmerga. The Federal Ministry of Defense said it will "make a situation-based decision whether to resume training," a spokesman said.
The operation puts the United States in a dilemma and highlights the complexities of the...
Read More: Kirkuk: US appeals for calm as allies clash in Iraq