Is it normal to fight players with lots of legendaries?

There are two major play modes outside of Arena, ranked and casual. In both you are supposed to be matched against players of similar strength, the difference is that ranked is reset every month and your ranking is visible.

The matchmaking doesn't work perfectly, one issue is that some players concede intentionally to drop their rank so that they can then do their quests against easy enemies. The other major issue is that in ranked the rank is reset every month, you play against much better opponents at the beginning of the month than at the end if you're at a low rank yourself.

Casual is generally less predictable, you might get an opponent that tries out a fun, but not really competitive deck, or you'll get someone grinding quests with a very good deck.

If you want to have fun playing Hearthstone, you shouldn't let it bother you that your opponents have better cards. They will have more legendaries than you very often, unless you pay a lot or have played for a long time.

Legendaries are not as important as it seems to a new player. Yes, they're better than the cards you have right now, and a player with legendaries currently has an unfair advantage over you. But the overall quality and composition of the deck matters more, with a few exceptions most of the competitive decks have rather few legendaries (but those legendaries are rather important).

As a general rule, more aggressive and faster decks are cheaper and easier to build for a new player. They usually don't need legendaries, decks like Warlock Zoo or Aggro Hunter only need a few rares to work. And if you win before turn 8, it doesn't matter that your opponent had the big stuff on his hand.

I also found it occasionally satisfying to beat a deck that had more legendaries than I have in my whole collection. I once played Mage against a Control Warrior, a competitive deck with a lot of legendaries. I anticipated that he had a lot of them on his hand, I got lucky and drew my sheep and a fireball, and just removed every legendary he played for four turns straight. While this doesn't happen very often, it was quite fun.