Rider's shout of joy suppressing a scream (10)
I believe the answer is:
charioteer
'rider's' is the definition.
(someone riding a chariot)
'shout of joy suppressing a scream' is the wordplay.
'shout of joy' becomes 'cheer' (a cheer is an expression of joy).
'suppressing' is an insertion indicator.
'scream' becomes 'riot' (synonyms).
'a'+'riot'='ariot'
'cheer' going around 'ariot' is 'CHARIOTEER'.
(Other definitions for charioteer that I've seen before include "Driver of Roman vehicle" , "Porphyrius perhaps" , "Her erotica (anag)" , "Driver of horse-drawn fighting vehicle" , "Hippodrome racer" .)