In all the discussions way back when with original Outpost about when a second/failed Outpost was supposed to be discarded, I don't ever remember it being said that Outpost #2 would get discarded from play on a different turn from Outpost #1. I always thought it was Outpost #2 gets discarded on your Outpost turn instead of on the current turn, because it didn't know yet that it wouldn't do anything (and maybe also because it hasn't made you draw 3 cards instead of 5 yet).
But with this understanding, it seems clear that original Outpost (same as brand new Outpost) would make it so that only the first Outpost gets discarded on the Outpost turn; the second Outpost gets discarded on the turn after that (opponent's turn).