Rise making popular Conservative with royal relax (8)
I believe the answer is:
increase
'rise' is the definition.
(I know that rise is a more specific form of increase)
'popular conservative with royal relax' is the wordplay.
'popular' becomes 'in' (something that's 'in' is fashionable).
'conservative' becomes 'C' (UK political abbreviation).
'with' says to put letters next to each other.
'royal' becomes 'r' (common abbreviation e.g. RM - Royal Marine).
'relax' becomes 'ease' (eg to ease a restriction is to relax it).
'in'+'c'+'r'+'ease'='INCREASE'
'making' acts as a link.
(Other definitions for increase that I've seen before include "Become greater in number" , "boost" , "extra money at work" , "Grow greater" , "Augment or enlarge (8)." .)