The 19-year-old has returned to his home country in a bid to get regular first-team football.
Fulham manager Martin Jol said: "Kerim has progressed well through his time in the Academy and into the fringes of the first team, which enabled him to play for the Turkish national team, which is a big thing for him.
"But for him to continue this progression and to really fulfill his potential he needs to play week in, week out, and I couldn't offer him that as part of the first team here, so we decided to accept the offer from Besiktas.
"He's sure to have a big impact back in Turkey and I know that on a personal level it's a good option for him."
Jol added on Fulham's official website: "As a club, because of the structure of the deal, we are still invested in his future and what we believe he'll be able to achieve in time. I know he'll be much happier playing, so we wish him all the best."