ISTANBUL—Turkey said it attacked Kurdish forces in Syria backed by the U.S., potentially setting the stage for a new fight with the Obama administration as it looks at increasing its ties with the Kurdish militants leading the most effective fight against Islamic State.

In a television interview Monday, Prime Minister Ahmet Davutoglu said Ankara carried out two strikes on the Kurdish YPG group as a warning to the fighters against expanding their control into parts of Syria that Turkish officials hope will serve as a safe zone...