They show how popular TV shows are with sailors (7)
I believe the answer is:
ratings
'sailors' is the definition.
(British term for naval enlisted men)
'they show how popular tv shows are' is the wordplay.
I cannot quite understand how this works, but
'popular' could be 'in' (something that's 'in' is fashionable) and 'in' is present in the answer.
'are' could be 'a' (short for 'are', historical unit of measurement) and 'a' is present in the answer.
This may be the basis of the clue (or it may be nonsense).
'with' is the link.
Can you help me to learn more?
(Other definitions for ratings that I've seen before include "Royal Navy sailors - staring (anag)" , "ordinary seamen" , "Information about people watching" , "they matter to broadcasters" , "Measure(s) of popularity" .)