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"X is normal" means "X is an accepted part of the social culture and norm set that you and I share." Comes with the assumed-shared worldview that (a) you can only fairly be blamed for violations of the norm set, and (b) all assertions of "badness" are this sort of blame.
i like the separating out of points a) and b)! but "you and I share" doesn't quite fit the usage here, i think. as addressed to the outgroup, the OP's message is more like "your norms are bad and should change"
Yes, but that's the intended meaning of the tweet, not the underlying, assumed-to-be-shared concept. I still think the underlying "normal" concept is roughly what I said, but lots of meaning is further added via implicature.
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