Bird that's top, not bottom (4)
I believe the answer is:
crow
'bird' is the definition.
(I know this)
'top not bottom' is the wordplay.
'top' becomes 'crown' (crown can mean the top of something).
'not bottom' means to remove the last letter (the letter at the bottom of the word is removed).
'crown' with its final letter taken away is 'CROW'.
'that's' acts as a link.
(Other definitions for crow that I've seen before include "Black bird; boast" , "Common bird" , "shout in triumph" , "Brag; carrion bird" , "Boast, swagger, triumph" .)