The Doctor is a little bit handsome
This is kind of cheating, as no English speaker would use that terminology. He is either handsome or not, and if he is in between, we would say he is "quite nice" or "alright".
A little bit handsome is almost pointlessly Shakespearian in its Olde English connotations - and it should NOT be taught like that on Duolingo.
I just assumed that the structure was possible in Turkish, and that we had to give a direct translation. Sometimes, we have to write sentences that are not correct English because we are translating a Turkish structure. This particular sentence didn't actually bother me. I think that there are levels of handsomeness. After all, it is perfectly acceptable to say that someone is very handsome, so it must also be grammatically acceptable to say that someone is a little handsome. In the right context, I might use "he is a little handsome" for someone who is not handsome, but who I don't think is ugly. I would never use "a little bit" with an adjective, but I don't speak American English. I assumed that it was acceptable in American English.