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I think it would be exactly like you said lolDepends on what you mean by "supernatural."
Bigfoot exists? Alright we found a really reclusive new species of Great Ape we somehow missed. See if we can get David Attenborough to voice the documentary, assuming we get more footage. World moves on.
Alien abductions have really happened? Nothing in science changes but HOLY **** ALIENS CONFIRMED, FERMI PARADOX SOLVED, the Internet breaks for a bit, Wikipedia's servers are stressed, but society is still functioning. Astronomy becomes a much more popular hobby though.
Angels / Sleep Demons / Magic / Ghosts / etc? Society in shambles, science is wrong, standard model thrown out the window, they faked the Moon Landing, Amaterasu can't melt steel beams, Dungeons and Dragons is mind controlling children, Mark Zuckerberg is a lizard confirmed, humid sacks of rice, dogs and cats living together, mass hysteria, etc.
My mind would rest easy at stuff like the first example, but anything more is a No.
Collapse is a strong word but the world would indeed be changed forever. Depending on what is discovered it may result in any sorts of changes but I doubt it would be overwhelmingly more impactful than the Renascence or the fall of of Rome, along with other "era changing" moments.If I discovered that magic is real and used by some hidden organizations, I would honestly die from paranoia lol. Discovering just a single supernatural element in our world would probably collapse human society if we think realistically.