Could it be translated as "straight to the hospital"?
Why isn't 'to the hospital'
Because they aren't walking "hastaneye" but only "hastaneye doğru".
I thought that i am a bird.
My bird and me... should be accepted
'Me' is not used as a subject. As a subject, 'I' is used. The subject is 'my bird and I'. - https://en.oxforddictionaries.com/usage/i-or-me