Green’s Dictionary of Slang

to act in an aggressive or extreme manner, to pursue aggressively.

1964		New Statesman 10 Apr. 555/2: ‘To strong it’ means to overdo something, like taking more than 30 purple hearts in one night.
1970		G.F. Newman Sir, You Bastard 97: Don’t you think that’s stronging it?
2000		J.J. Connolly Layer Cake 87: The secret, he told me once, is not to strong it.
2007		N. ‘Razor’ Smith Raiders 83: It wasn’t a bad prison [...] the screws didn’t strong it with anyone. [Ibid.] 85: I couldn’t strong my application in case I got the screw suspicious.