Yes, this is normal. You can use me as another example! Puzzle ratings do not have an ELO system like actual chess. Rather, the rating system is a way for the site to give appropriately rated puzzles to you (not too easy and not too difficult).
Why is my rating so bad
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Not sure what a good puzzle rating is, mine is close to 2000 but my blitz rating is hovering around 1000. Is it normal to have such a huge gap between puzzles and blitz?
Yes, it is normal.
Puzzle ratings really aren't like the competitive play ratings. Persistance really pays off in puzzle ratings. I see you have done 7216 puzzles. That's a pretty high number.
Hopefully, all the work you put in on puzzles will be useful to you in the long run and your play ratings will rise.