Monopoly Deal: Fairest way of removing extra cards
There are two options:

The rules as written say that they should go on the bottom of the deck.
You could opt to have players put extra cards face up in the public discard instead.

The official rulebook says the following (my emphasis):

If you have more than 7 cards in your hand at the end of your turn (not including cards on the table) discard extras to the bottom of the draw pile so you only have 7.

This is an unusual way for games to handle discards; I think that this is an inelegant solution, and it sounds like you do too.  monopolydealrules.com is non-authoritative collection of rules clarifications and improvements sourced from the community, and suggests a discarding face-up to the discard pile.  Their &quot;general rules&quot; say:

At the end of a turn, a player cannot have more than 7 cards in their hand. If they do, they need to discard the excess cards into the discard pile in the middle.

This is only a meaningful difference if you actually go through the entire deck in the game, which is unlikely.  I'd recommend going with having discards go to the discard pile, as it's a more normal way to handle discards and it addresses the concern of players potentially manipulating the deck through discards.