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 really see how this works, but
'popular' could be 'in' ('in' can mean trendy) and 'in' is found within the answer.
'are' could be 'a' (short for 'are', historical unit of measurement) and 'a' is located in the answer.
This explanation may well be incorrect...
'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" .)