Jesus! What an infamous game show host (5,7)

I believe the answer is:

chris tarrant

'game show host' is the definition.
(Who Wants to Be a Millionaire host)

'jesus what' is the wordplay.
'jesus' becomes 'christ' ('Christ' can be a synonym of 'jesus').
'what' becomes 'arrant' (I can't justify this - if you can you should give a lot more credence to this answer).
'christ'+'arrant'='CHRIS TARRANT'

'an infamous' acts as a link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.

I've seen this clue in The Guardian.

Want a hint initially instead of a full solution? Install my app