3 misconceptions here, but #3 shows that you're on to something:
- There are other factors besides puzzles that influence rating. Even if chess is 99% tactics (In this case tactics is synonymous to puzzles?), I believe the pareto principle has a lot to say for that 1%. There's still eg endgames.
1.1. In particular, I bet you're talking more about chess not chess960. So openings are relevant too.
- Even if there's a relation between puzzle rating and chess rating, it doesn't mean there's an exact relation. It could be that the 2000 puzzle rating converts to 1800 rapid. For example this rating converter https://www.chessratingcomparison.com/graphs says 2000 puzzle is about 1378 in rapid.
Look at even the plus minus 187 and the scatter plot. I don't think that's even like 50% correlation. ( But then again I was never the best guessthecorrelation player.)
- There's even a (significant) relation between untimed puzzle rating and chess rating. I say more here: https://chess.stackexchange.com/questions/41619/is-my-chesstempo-timed-puzzle-rating-actually-expected-to-relate-to-my-chess-rat
People could get the same puzzle correct but at largely different times. Additionally, not everyone is doing the untimed puzzles the same way: People could take their time like I do or set an arbitrary time limit for themselves, eg 5min per puzzle. Thus, there's not really an expectation that there's gonna be that much of a relation between untimed puzzle ratings and regular chess ratings. I believe in theory an untitled player could even reach the same level as even a titled player in puzzle rating but still have a huge gap in chess (or chess960) ratings.
This is where chesstempo's blitz mode comes in: This feature is so beautiful and incredibly underrated in that it's so far the only thing I've seen of its kind in the history of chess. You're now graded based on your time relative to the time others have taken. Even if there's an inherent (yet unspecified) time limit of around 1 minute, it's still a 'reasonable' time limit in the sense that it's not arbitrary: The time limit is not the same for each puzzle and is based on your rating, the puzzle's rating and other players' ratings that you're expected to complete the puzzle in within the time limit. This is opposed to doing untimed puzzles where you might set an arbitrary time limit on yourself and as opposed to a hypothetical site that imposes, say, a 5 minute time limit on every puzzle (I think the hypothetical site is better than arbitrary self-imposed time limits but not as good as chesstempo).
Note on uniqueness: The closest I've seen to chesstempo's blitz mode is https://blitztactics.com/ which gets from lichess' puzzles but afaik the puzzles don't necessarily change ratings depending on the lichess' puzzle or player ratings.
Note on lichess: lichess' puzzle storm or puzzle racer aren't really like chesstempo's blitz mod because the puzzles and players don't really have ratings.
So what I think you're looking for (and what would be better for chessratingcomparison.com to show) is a conversion between chesstempo's timed puzzle ratings and your chess ratings. I think it'd be similar even to like a conversion between ratings between any 2 sites say chesscom rapid and lichess rapid.
In chessratingcomparison.com, the scatter plot definitely shows high or at least much higher correlation compared to the puzzle rating to chess rating above:
In fact, I don't think you even need chessratingcomparison.com for this. As long as you're playing regularly and doing puzzles regularly ('regular' is relative to the rating deviations of each rating system I guess), you should be able to get a feel that eg 1550 in chesstempo converts to about 1600 in chesscom rapid especially if after awhile, say, there's a period of the last 3 months where your rapid rating is about 1520-1589 while your chesstempo timed puzzle rating is about 1470-1539.
Re
How do I bridge that gap and increase my game score?
If your timed puzzle rating is about 1550 (according to the aforementioned example I gave), then I guess there's not really a 'gap' to bridge.
As to what happens if there really is a gap? Well I have a feeling you'll gain enough rating with some more games if your timed puzzle rating really says so.
Bonus: I think that this conversion is even more accurate in chess960 compared to chess since there's 1 less factor involved, namely openings.