Bird that's new coming up over river (4)
'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.
'over' indicates putting letters inside.
'river' becomes 'r'.
'new' written backwards gives 'wen'.
'wen' placed 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", "Architect of St Paul's Cathedral", "Small bird (in the Navy?)".)