I am pretty new to chess, and I play lot of chess puzzles: trying to find the best move for a given situation (which will not necessarily end-up with a checkmate).
Is this a good way to learn chess or should I play more normal games?
Solving alot of tactic puzzles is a fantastic way to improve for a beginner.
One of the reasons for my improvement was the constant solving of tactics on chess.com's tactics trainer feature, which basically made me shed my beginner skins, since tactics almost always play a decisive role in the games of beginners.
However, you must also play many normal games, preferably in real life instead of online, and against stronger players, as this will provide you with invaluable experience for the improvement of your game.
You must also familiarize yourself with the strategic features of chess, (backward pawns, space advantages) and the endgame in order to improve. Tactics can only get you so far.
Is [solving chess puzzles] a good way to learn chess or should I play more normal games?
This gets the classic answer for an either/or choice that needn't be exclusive: yes. In other words, do both, as these are two of the surest ways to improve, and each is all but necessary in order to do so. Of course, one's time is limited, and so your question may rightly be: which to focus on more. That should probably be decided by an analysis of your own games, especially losses. For instance, if you can trace most losses to tactical oversights that you wouldn't have recognized were possibilities before they happened, then more tactical training is definitely in order. On the other hand, if you find yourself committing mistakes where you should have known better, but lost focus, say, then more time playing games might be needed. But I want to reiterate the case for finding some time for both.
It's an empirical fact that tactical mistakes (and taking advantage of tactical opportunities that are presented) tend to play a very decisive role in chess, and that fact is amplified at the lower levels (and on up to pretty high levels even); after all, it's hard to come back after hanging your queen. Beginner players in particular will tend to make very egregious, serious-material-losing mistakes quite often, and so simply cutting down on such oversights (and learning to spot when your opponent makes them) can pay enormous dividends in one's skill/results rather quickly. Solving tactical chess puzzles (e.g. at ChessTempo) is the best way to hone one's skill in this area, and can lead to very rapid improvement.
That said, merely doing that can only get one so far. Playing actual games from start to finish provides invaluable experience for ... playing actual games from start to finish. You could spend countless hours in a gym shooting thousands of shots from every spot on a basketball court, to the point where you are an excellent shot. That would undoubtedly make you a more skillful basketball player. But it wouldn't prepare you to struggle against stubborn defenders, or to withstand the rigors of running up and down the court for a long period of time, or to sense where you should position yourself on the court in the flow of a game. Similarly, solving tactics puzzles doesn't prepare you for the give-and-take struggle for initiative and advantage in a chess game, or offer practice in turning a hard-earned advantage into an actual victory (or setting up stiff resistance when things haven't gone your way), or maintaining your focus throughout the entire course of a game. To progress in terms of competitive results pretty much demands playing more actual games as well.
Definitely not a bad approach but one thing you should know is the importance of strategic/positonal play cannot be emphasised enough.
Tactics can only get you so far. If you're tactically sound but your positional play is wanting, that will pose trouble.
Against other beginners, sure you will win a couple games but as you move up the ladder, things are bound to get "interesting"
It's a decent way. But there are some problems with it. Sometimes the puzzle's AI moves their piece even though another piece could be moved better. But this way you can practice how in some situations you can win a piece (like winning a Queen) or check-mate the opponent. But to really become good in playing chess, you have to play normal matches as situations you come across in normal matches are not always in puzzles and you need to know how to win / prevent losing.
Good luck and I suggest you keep on training, but not only training this way, but also play normal chess.
I suggest playing normal chess against the same rating, but not AI's. AI's can be a little 'dumb' sometimes. I noticed it a few times on lichess.
I agree with other comments but I also believe you should have an approach to studying tactics such as recommended by Alexander Kotov in "Think Like a Grandmaster" (I skipped a rating class after studying this book) or in the book "Imagination in Chess" That is, it helps to have an understanding of tactics and a systematic way of studying them.