(CNN) Iraqi security forces took direct control of the strategic city of Kirkuk, driving hundreds of families from the oil-rich territory that had been under Kurdish control for more than two years, security sources and witnesses told CNN.

The forces entered the disputed city on Monday and set up checkpoints around its perimeter, while witnesses saw a sole Iraqi flag flying atop the governor's headquarters. The building usually holds both the Kurdish and Iraqi flags.

The breach of the city came after Iraqi forces pushed through two fronts on Kirkuk's outskirts, seizing key oil fields and a military base, among other valuable assets. Iraqi counterterrorism forces earlier Monday said they would not enter the city proper.

The operation was ordered by Iraqi Prime Minister Haider al-Abadi, who sent federal forces to the region to impose security and called on Kurdish Peshmerga forces to cooperate with them. The majority of local Kirkuk police have remained and are working with federal officers.

His order came just weeks after a Kurdish referendum for independence claimed Kirkuk as their own, causing consternation in Baghdad. The city -- among the most coveted of disputed territories in the region -- falls outside the internationally recognized autonomous Kurdish region in the northwest of Iraq.

The Iraqi operation is supported by the Popular Mobilization Units, an Iranian-backed militia group that has helped spread Tehran's influence in the Middle East.

Clashes between the two sides erupted overnight. At least 16 Kurdish fighters were killed and dozens were wounded in confrontations with Iraqi forces, two senior Kurdish Peshmerga commanders told CNN.

A dilemma for the US

The Iraqi operation, which began late Sunday, was the clearest indication yet of Baghdad's determination to curtail the territorial ambitions of the Kurds.

It pits two key US allies against each other, highlighting the complexities for Washington in the fight against ISIS -- the US arms and supports both the Iraqi forces and Kurdish fighters. It again raises the long unanswered questions of how territory might eventually be divided along ethnic lines.

The US State Department released a statement Monday night saying the US "is very concerned by reports of violence around Kirkuk," and Washington is "monitoring the situation closely."

"We strongly urge all parties to avoid provocations that can be exploited by Iraq's enemies who are interested in fueling ethnic and sectarian conflict," the statement said. "In particular, we note that there is still much work to be done to defeat ISIS in Iraq, and continued tensions between Iraqi and Kurdish forces distract from this vital mission."

The US-led coalition had earlier played down the overnight clashes, saying in a statement it was "aware of reports of a limited exchange of fire" in darkness before dawn, saying it appeared to be "a misunderstanding and not deliberate as two elements attempted to link up under limited visibility conditions."

The statement made clear that it was not supporting either side near Kirkuk.

The Kurdistan Regional Security Council said Kurdish fighters were attacked using "US military equipment, including Abrams tanks and Humvees." It added the Peshmerga had destroyed a number of those Humvees.

In an interview with CNN, Bayan Sami Rahman, the representative of the Kurdistan Regional Government to the United States, blamed the Iranian-backed militia for the fighting overnight. But she still called for dialogue with Baghdad.

"Our offer of dialogue with an open agenda remains on the table," she told CNN on Monday. "We urge our friends to use (their) leadership role to prevent war."

A contentious city

The greater Kirkuk province has one of the biggest oil fields in the country -- more than 6% of the world's oil comes from the area.

By Monday, Iraqi troops had taken control of several oil fields, including the key Baba Karkar oil and gas field and the K1 military base, among other assets, Iraqi counterterrorism spokesman Sabah al-Noman told CNN.

Iraqi forces fled Kirkuk in 2014 as ISIS fighters attempted to secure the territory, shortly after taking over the city of Mosul and establishing their so-called Islamic caliphate across the northwest of the country.

The Kurds, however, sent in their fighters and claimed the city.

Kirkuk was historically a Kurdish-majority Iraqi town, but during his rule, ousted dictator Saddam Hussein moved Arab families in and Kurdish families out to change the area's ethnography, under a policy termed "Arabization." It's also home to Sunni Arabs and Turkmen.

The city has suffered a series of major attacks over the past decade from extremists, including al Qaeda in Iraq, targeting mostly security forces there.

After the fall of Saddam, Kurds began returning to Kirkuk, repopulating the city and its surrounding areas in the event of an eventual referendum on whether the city should be part of a future Kurdistan or remain in Iraq.