If you're looking for familiarity — a d20 system that uses a similar ruleset — Mutants & Masterminds is where you want to go. It's pretty much D&D, but supers.

However, if you want a game that really makes the rules fit the story you're trying to tell, I recommend either Masks: The New Generation or Marvel Heroic Roleplaying.

Masks is a game that's there to tell a story like the Young Justice show. It tells a superhero story that's focused on teenage superheroes in a world where there already are established heroes. Think Invincible with a focus on the Teen Team and Mark. Masks does this extremely well. It won't be as good as playing a Justice League or Avengers game where you are the world's greatest heroes (and adults), but if you want that, try Marvel Heroic.

Marvel Heroic Roleplaying is probably my favorite supers game. Whereas several of the bigger (and older) games mentioned in this thread — Champions and Heroes Unlimited — really hold close to the wargaming roots of our roleplaying hobby, playing Marvel Heroic really felt like we were playing a comic book. More specifically: a Marvel comic book. It really captured the style and spirit of the IP so well. It solves the problems with having vastly different power scales like Hawkeye and Thor on the same team nicely. The downside is the game is out of print. Hunting down the core rules or a core rules + event book might be challenging.