"ok, imagine a gun."
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It means "quick marriage because the bride is pregnant" and that is 100% the origin of the phrase.
Particularly in poorer, rural parts of the USA having a child out of wedlock was incredibly shameful, and the financial burden of a single motherhood was intolerable. So the bride's family would ensure the man responsible married their daughter ... regardless of how he felt about it. Sometimes that meant having a shotgun at the wedding to ensure he didn't run off.
After thw wedding is done the father goes "And kid you'd be no smoking anymore get me? I don't wanna hear you went to buy a pack or whatever..."
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That is purely an American thing.
Not saying my family had someone in the passenger seat with a shotgun to protect their batch of white lightning...also not saying they didn't.
we have been using this in Poland at least for 10 years