I looked at the game in question, and the following happened (copied from the log, "I" means what you did). This is a good example of some rules intricacies:
I plays an Overlord.
I plays a Lighthouse.
I gets +1 Action.
I gets +$1.
I calls a Royal Carriage.
I plays an Overlord again.
I plays a Monument using Way of the Turtle.
Way of the Turtle fails to set aside a Monument.
The most important thing to know about this situation is that Overlord (and similar cards, like Band of Misfits) got a wording change roughly 2 years ago.
The old wording made the Overlord "become" the other card as long as it was in play, but that's no longer true. The new wording (used in the online game) is
Overlord: Action - Command, 8D
Play a non-Command Action card from the Supply costing up to $5, leaving it there.
The new wording makes the Overlord never "change form", it stays an Overlord, but instead it plays the "chosen" card from the Supply, but leaves it there, it's not moving into play. That means you don't gain the card or otherwise add it to your deck, which is consistent with the old behavior.
OK, first you play Lighthouse with Overlord:
I plays an Overlord.
I plays a Lighthouse.
I gets +1 Action.
I gets +$1.
That's the normal things you get when you play Lighthouse. As I said, the Lighthouse does not move into play. As the new Overlord text says, it stays in the Supply. If you look at your game board at this time, the Lighthouse you see in play is the one from your
previous turn, not the one just played. If it was the one just played, it would be to the right of the Overlord:

That has an immediate consequence: When the "old" Lighthouse is discarded from play during Clean-Up this turn, you are no longer protected from attacks on your opponent's turn. Lighthouse reads, under the dividing line:
While you have this in play, when another player plays an Attack card, it doesn't affect you.
After his turn, you don't have a Lighthouse in play, so you are not protected!
So the turn continues:
I calls a Royal Carriage.
I plays an Overlord again.
I plays a Monument using Way of the Turtle.
Way of the Turtle fails to set aside a Monument.From what we saw above, the Monument played by Overlord is
not moved into play. That means Way of the Turtle can't find it. It expects a played card to be in play, which the Monument isn't. That results in the bolded message above: The Monument is not set aside.
Now look at the text of Way of the Turtle:
Set this aside. If you did, play it at the start of your next turn.
The Way of the Turtle did not set the Monument aside, so it's isn't played next turn, just nothing happens because of this.
What you saw next turn, getting the +$1 was from the
Lighthouse you played with Overlord!
What could you have done?
When you replay the Overlord with Royal Carriage, play the Overlord using Way of the Turtle! The Overlord is in play, so can be set aside, and next turn you can then decide what to play with it (can be Monument, or something else that seems reasonable next turn).