Shove up, turning quiet (4)
I believe the answer is:
push
'shove' is the definition.
(I know that shove is a is a more specific form of the action push)
'up turning quiet' is the wordplay.
'turning' says the letters should be written backwards.
'quiet' becomes 'sh' ('sh' is used to tell someone to be quiet).
'up' reversed gives 'pu'.
'pu'+'sh'='PUSH'
(Other definitions for push that I've seen before include "What often proverbially comes to "shove"?" , "Press forward" , "Force one's way (through)" , "Strongly promote" , "Offensive - press" .)