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human
[Godzilla] How did Godzillasaurus survive the K-Pg mass extinction?

Godzillasaurus was the ancestor of Godzilla, but unlike him, he just appears to be a regular theropod dinosaur.
gpt
Unfortunately, we don't know a lot. The world's governments are more interested in researching the current, mutated Godzillasaur that we know as Godzilla, than in allowing expeditions to find more about the pre-mutation Godzillasaur. 

The only documented encounters we have with a specimen of Godzillasaurus, prior to the appearance of Godzilla is from WWII, on the Pacific island known as Lagos (not to be confused with the one off Africa), located somewhere near the Gilbert and Marshall Islands. Eyewitness accounts are spotty, and the veterans are mostly gone these days. Plus, G-Force has denied most civilian research on the island for fear of Godzilla returning there on one of his rampages.

There is not a lot of publically-available geographical and ecological data on the island, or how old it is geologically. The islands in that part of the world are volcanic in nature and may likely be younger than the K-T Extinction.

Likewise, there has been no fossil evidence of Godzillasaur found (not surprising, as there are likely millions of extinct species we've never found fossil evidence of). We don't know what their native range was or what they evolved from. Despite some resemblance to therapods, Godzillasaurus apparently had four toes. Some speculate that the species did not evolve until well *after* KT. This hypothesis states that a small, warm-blooded carnivorous dinosaur species, perhaps with some amphibious traits, survived KT along with small mammals, and somehow found a niche to occupy in the post-KT world. Eventually this species's evolutionary descendants migrated toward the Pacific and found its way to Lagos before dying out elsewhere. On Lagos, the Godzillasaurus evolved through convergent evolution to become the island's apex predator, with strong resemblance to the therapods of old.

From there, nuclear tests had a strange effect on the last surviving (as far as we know) Godzillasaurus's DNA, causing the mutation into Godzilla.

(Note: This is based off the canon of the Heisei-era Godzilla series, beginning with Godzilla 1984 and ending with Godzilla vs. Destroyah)