Popular island facing hill — place of growing culture (9)
I believe the answer is:
incubator
'place of growing culture' is the definition.
(I've seen this before)
'popular island facing hill' is the wordplay.
'popular' becomes 'in' ('in' can mean trendy).
'island' becomes 'Cuba' (Cuba is an example).
'facing' is a charade indicator (letters next to each other).
'hill' becomes 'tor' (tor is a kind of hill).
'in'+'cuba'+'tor'='INCUBATOR'
(Other definitions for incubator that I've seen before include "what will help early life?" , "Egg-hatching apparatus" , "hospital apparatus" , "Promotor of growth" , "Egg hatching apparatus" .)