It is called devoicing. When we pronounce an R in English (or in say Spanish, which has the similar alveolar flap), it is voiced, meaning your vocal cords vibrate. In Turkish, R are the end of a word is devoiced, meaning your tongue moves and air comes out, but vocal cords stay still. To the untrained ear, it does sound a lot like a sh, but there really is an R sound there.
Yes, just to play devil's advocate here, this is a sentence that could come straight out of a fairy tale, with talking mice, etc. Tavşan is one of the characters, and so his name translated into English is Rabbit. (To bolster my case, there's a capital T here, after all.) Think of all the Native American (American Indian) stories with the characters Wolf, Bear, Fox, etc. If they're going to give us a sentence about a rabbit that reads, why can't we imagine it's a character named Rabbit?
Referring to something specific is definite, which is independent of being accusative. Referring to what the action is being done to, the (direct) object of the verb, is accusative. The so-called accusative suffix in Turkish is really a definite accusative suffix; if you don't have both if these, then you don't use the suffix.
Anyway, both nouns here are definite, but only the second one is accusative.