Bet sketch shows old American dandy (6,6)
I believe the answer is:
yankee doodle
'old american dandy' is the definition.
(as in the song I'm a Yankee Doodle Dandy)
'bet sketch' is the wordplay.
'bet' becomes 'yankee' (type of bet in gambling).
'sketch' becomes 'doodle' (I have seen 'Rough sketch ' mean 'doodle' so perhaps 'sketch' could also mean 'doodle').
'yankee'+'doodle'='YANKEE-DOODLE'
'shows' is the link.
(Other definitions for yankee doodle that I've seen before include "Who came to town, riding on a pony" , "Traditional US song" , "American national song, popular since the War of Independence" , "American marching song" , "Old US popular song" .)