Supporting book, European, in thorough enquiry (5)
I believe the answer is:
probe
'thorough enquiry' is the definition.
(I know that probe can be exhaustive)
'supporting book european' is the wordplay.
'supporting' becomes 'pro' (being pro something is supporting it).
'book' becomes 'b' (abbreviation for book).
'european' becomes 'E' (abbreviation e.g. EU).
'pro'+'b'+'e'='PROBE'
'in' acts as a link.
(Other definitions for probe that I've seen before include "(Crime) investigation" , "Examine searchingly" , "Closely enquire into" , "Question or examine thoroughly" , "Poke into" .)