That's very odd, that they do it right away.

The purpose of the rule is to keep the hand unaffected as much as possible.

You kill the exposed turn card, then burn and turn the "river," because even with the mistake, that would have been the river the whole time. Then you reshuffle and turn another card without burning. That way, the second burn card stays, and the only card in the deck that is different from its original intention is that final card, and even then it has a chance of coming up the same as it did when it was the original 'turn.'