No, it's not rare at all. In fact, I find "newborn baby" to be redundant. You can have babies that are not newborns, but you can't have a newborn that is not a baby.
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Grammar GeekIn fact, I find "newborn baby" to be redundant.I don't agree. As you say,
Grammar Geek It's just "newborn baby" alone that is redundant.That's your opinion, but there are 220 citations for 'newborn baby' in COCA, so quite a few people feel differently.
Grammar GeekBut we do use it as a noun fairly often.We don't in BrE.