We've taken ours on the dolmus from Olu to Hisaronu after a walk with the Kismet group. Most of the passengers were English and didnt bat an eyelid.
Another time we went from Yaniklar back to our car after a walk and it was full of villagers, so we sat at the back with Honey safely by our feet and no one took any notice.
We often get tooted to see if we want to get in when we've got Honey with us, so assume it's ok but have always felt it isn't...
So, maybe it's down to the individual driver, or maybe it's ok but just something Turkish people wouldn't do as their dogs generally live in the garden.
My worry is more that a lot of Turkish people are scared of dogs, so it just doesn't feel very fair.
As it's a puppy I would be inclined to carry it and keep it away from the other passengers and don't think there will be a problem.
Going to Ovacik will be mostly tourists who will probably all just want to play with the puppy.