American police department is sharp (4)
I believe the answer is:
acid
'sharp' is the definition.
(I've seen this before)
'american police department' is the wordplay.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'police department' becomes 'cid' (I've seen this in other clues).
'a'+'cid'='ACID'
'is' is the link.
(Other definitions for acid that I've seen before include "Sour - LSD" , "Substance which turns litmus red" , "Caustic liquid of many kinds" , "- - -drop, sweet" , "Corrosive liquid; drug" .)