Power Grid gives a turn order advantage to players who make less income.

Dominion is a deckbuilding game where you earn victory points by buying and keeping junk cards in your deck, so your deck gets worse as you score increases.

Suburbia reduces your income progressively as your population (score) increases (which also forces players to balance the two).

A number of games including Terra Mystica give turn order advantage to whoever passed their turn first in the previous round.

There are, of course, many more games that don't have such a mechanic, but rely on balancing the relative rewards of the various moves such that early turns are less about taking a lead and more about setting up which paths to victory you can attempt in later rounds, so the skill is in setting up a path that you're able to execute well and then executing it well.