lol! Ark encounter is suing its' insurance company

[Bernardo de La Paz]

23. It is not correct.

There are two cases for "its" and the apostrophe does not come after.

1) "it's": the contraction for "it is".

2) "its": the contraction for possession. I remember this by saying to myself that the 's' is closer to the 'it'. The newspaper article got it right.

There are two situations, not related to "it's" where the apostrophe comes after.

A) Where the possessing noun is plural. Ex.: "The companies' industry sector failed."

B) Where the noun ends in 's'. Ex.: "Jesus' socialism does not have Republican approval."

Neither of those apply to the OP.