Horse in street starts to exasperate every driver (5)
I believe the answer is:
steed
'horse' is the definition.
(steed is a kind of horse)
'street starts to exasperate every driver' is the wordplay.
'street' becomes 'st'.
'starts to' indicates taking the first letters.
The initial letters of 'exasperate every driver' is 'eed'.
'st'+'eed'='STEED'
'in' is the link.
(Other definitions for steed that I've seen before include "Stallion" , "Powerful horse" , "(Poetic) horse" , "Swift horse" , "A mount (poet.)" .)