"The bird is next to the cat."
Translation:Kuş kedinin yanında.
No; using the possessive suffix -u would give it the meaning "His bird is next to the cat" (or "her bird").
And if you were trying to use accusative -u: remember that that is only for definite direct objects. "The bird" is definite, but it's not an object; instead, it's the subject of the sentence. ("Who or what is next to the cat? The bird is.")