On a rainy May day, Turkish Prime Minister Recep Tayyip Erdogan led two of his closest advisers into the Oval Office for what both sides knew would be a difficult meeting.

It was the first face-to-face between Mr. Erdogan and President Barack Obama in almost a year. Mr. Obama delivered what U.S. officials describe as an unusually blunt message: The U.S. believed Turkey was letting arms and fighters flow into Syria indiscriminately and sometimes to the wrong rebels, including anti-Western jihadists.

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