Heading west, not even admitting making a mistake, shows bravado (7-2)
I believe the answer is:
derring-do
'bravado' is the definition.
(thesaurus)
'heading west not even admitting making a mistake' is the wordplay.
'heading west' says the letters should be written backwards.
'not even' becomes 'odd' ('odd' is the opposite to 'even').
'admitting' means one lot of letters goes inside another.
'making a mistake' becomes 'erring' ('err' can be a synonym of 'make a mistake').
'odd' reversed gives 'ddo'.
'ddo' placed around 'erring' is 'DERRING-DO'.
'shows' is the link.
(Other definitions for derring-do that I've seen before include "heroic deeds" , "Literary word for heroic courage or action" , "Brave acts" , "Daring deeds" , "'bold action'" .)