I'm interested in _____;
____ ile ilgilenirim. / Türk kültürü ile ilgilenirim.
I'm interesting in _____;
____ ile ilgileniyorum. / Türk kültürü ile ilgileniyorum. (I guess we mostly use this one, but both are true.)
___ interests me;
___ (benim) ilgimi çeker.
or
___ (benim) ilgimi çekiyor. (this one sounds more correct again, but both are true.)
*Using (Benim) is optional, because in Turkish language you can understand who is making the act by looking at the verb supplement. If you need more information about this, you can ask me :)
Hope this helps!)