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' says the letters should be written in reverse (in a down clue, letters go up).
'over' is an insertion indicator.
'river' becomes 'r' (abbreviation for river).
'new' reversed gives 'wen'.
'wen' going around '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?)" .)