Twist organ a bird uses to fly across river (5)
I believe the answer is:
wring
'twist' is the definition.
(wringing is a kind of twisting)
'organ a bird uses to fly across river' is the wordplay.
'organ a bird uses to fly' becomes 'wing'.
'across' is an insertion indicator.
'river' becomes 'r'.
'wing' enclosing 'r' is 'WRING'.
(Other definitions for wring that I've seen before include "Squeeze liquid from" , "Squeeze water out of" , "Twist and compress" , "Extract with difficulty" , "Squeeze moisture (from)" .)