Stick around river to see bird (5)
I believe the answer is:
crane
'bird' is the definition.
(crane is a kind of bird)
'stick around river' is the wordplay.
'stick' becomes 'cane' (cane is a kind of stick).
'around' means one lot of letters goes inside another.
'river' becomes 'r'.
'cane' going around 'r' is 'CRANE'.
'to see' acts as a link.
(Other definitions for crane that I've seen before include "American writer" , "Hoist; wading bird" , "Device for picking up" , "Wading bird - lifting device" , "Lifting structure" .)