Bird that's new coming up over river (4)
I believe the answer is:
wren
'bird' is the definition.
(wren is a kind of bird)
'new coming up over river' is the wordplay.
'coming up' is a reversal indicator (in a down clue, letters go up).
'over' indicates putting letters inside.
'river' becomes 'r' (abbreviation).
'new' back-to-front is 'wen'.
'wen' enclosing 'r' is 'WREN'.
'that's' is the link.
(Other definitions for wren that I've seen before include "Type of small bird" , "She served" , "Member of the former Women's Royal Naval Service" , "English church architect, d. 1723" , "Small bird (in the Navy?)" .)