“To smuggle in a West Virginia reference, [Jerry] West seems here like both the Hatfields and McCoys. He shoots himself repeatedly in the head, feet and private bits.” –Dwight Garner, New York Times, November 3, 2011
“To smuggle in a West Virginia reference, [Jerry] West seems here like both the Hatfields and McCoys. He shoots himself repeatedly in the head, feet and private bits.” –Dwight Garner, New York Times, November 3, 2011
notoneoffbritishisms.com 
To my ear, this sounds like one of those crude literal translations. “Bits” suggests disintegration of pieces (e.g. smashed to bits) small components which easily can be separated from the whole (e.g. the bits or cutting pieces of a hand-drill).
I and every Eastpondian I know would say “private parts”.
I have heard expressions such as “Mind me bits!” used in England to admonish a clumsy person endangering ones private parts. But my experience says that it’s highly informal, used only by “common” people. Never noticed “private bits”.