President Donald Trump announced in December that the US would withdraw the estimated 2,000 troops it has stationed in Syria because the fight against the Islamic State group (ISIS) had been completed.
But there are no easy answers.
In announcing the move, Trump said the 2,000 American soldiers leading the coalition against the Islamic State jihadist group while helping thwart an Iranian military foothold in Syria would be pulled out soon.
"If they are terrorists, we will do what is necessary no matter where they come from", Erdogan said.
European nations have been reluctant to take back citizens with ties to the Islamic State, not wanting the legal challenge of prosecuting them or the potential security risk if they are released.
"The president's decision to withdraw our folks from Syria in no way impacts our capacity to deliver on that", said the secretary of state, who later also met Jordan's King Abdullah II.
"It's one thing for the government to be very confident that an individual joined or tried to join ISIS".
"It's a bipartisan failure", said Mark Dubowitz, the chief executive of the Foundation for Defense of Democracies, a hawkish Washington-based think tank.
Meanwhile, the prisoner problem is only growing worse.
In a statement, the YPG identified the 16-year-old American teenager as Soulay Noah Su.
Pompeo's visit began as U.S. National Security Adviser John Bolton was set to depart Turkey without meeting with President Recep Tayyip Erdogan - an apparent snub over disagreements about the Kurdish fighters in Syria.
There are fears that the USA withdrawal will leave a door open for Turkey to assault the US -allied SDF fighters.
The PKK is considered a terror organisation by the US, EU and Turkey.
What is the US's approach to Syria?
A spokesman for the secretary said the two men discussed "the US commitment to Iraq's sovereignty and territorial integrity", as well as the new Iraqi government's "efforts to deliver stability, security and prosperity to all Iraqis". Both officials spoke on condition of anonymity because they weren't authorized to disclose the information publicly.
The U.S. State Department had billed the speech at the American University in Cairo as the most complete remarks yet on U.S. priorities in the Mideast by Pompeo, who is on an eight-day tour of the region. While that decision's timing is unclear, it is widely seen as abandoning the region and benefiting US rivals Russian Federation and Iran.
But Senate Democrats have said many nominees Trump puts forward are wholly unqualified to be ambassadors, or have backgrounds that should preclude them from confirmation-such as Christine Toretti, the ambassador nominee for Malta, who had a restraining order filed against her for "placing a bullet-riddled target sheet" in the office of her ex-husband's doctor. Islamic State, he said, was "defeated militarily but mission is not accomplished". For Trump and his minions to achieve more, they must provide some clarity of objective, especially consistency at the outset.
In Qatar, Pompeo is likely to find a somewhat skeptical reception from a regime that has been under relentless attack from Saudi Arabia and its other Gulf allies for its willingness to defend Iran.
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