Generally speaking, it's been my experience that when it feels like "very" rapid fire, it means that I'm trying to shoot faster than my ability will allow me to.
On the other hand, shooting within my ability never seems to feel really fast even when the timer says otherwise.
I guess what I'm trying to say is that it takes time to progress from shooting slowly and accurately to shooting rapidly and accurately and that progress is best made by trying to eliminate wasted time as opposed to trying to go fast.
From a practical standpoint, what does that mean? It means a lot of things, but here are a few to work on.
You don't need to stare a hole into the paper over the front sight. Once you see a good sight picture, you're good to go--let the shot break. If it doesn't hit where the sights are pointing then you yanked the trigger and you need to work on trigger control.
Don't waste time blinking and don't lose track of the front sight. If you have to open your eyes after every shot and reacquire the front sight, you're wasting time that is better used lining up the front sight. If you're not seeing muzzle flash for each shot then you're blinking and that's wasting time.
You can shoot or you can score, but you can't do both well. While you're shooting, SHOOT. Make sure you're working your fundamentals properly and the on-target results will take care of themselves. Resist the temptation to count hits or keep score between shots. When you're done shooting THEN you can score.
You need to be able to maintain your grip through the entire shot string. If you find that you're having to adjust your grip between shots, or every other shot, then you need to work on your grip, or look for a handgun/cartridge system that fits you better.
You shouldn't be having to work to line up the sights for followup shots. If the gun isn't settling back into a decent sight picture after recoil then you need to work on controlling the gun. You can't keep it from moving, but you can adjust your grip and stance so that the recoil causes the gun to move consistently and settle back after recoil so that it's more or less in the same position it was before the shot.
Don't let your trigger finger get lazy. Don't RUSH, but try to keep your trigger finger in motion. It should always be either pulling or releasing the trigger.
Don't let the gun push you around. If you're finding that the cumulative effect of recoil during a shot string is pushing you back or changing your stance, then you need to work on a more aggressive stance. Lean forward a little bit, position your weight a little more on your forward foot and keep your knees bent a little bit.
In other words, you want to make your shooting more efficient, more smooth. That will pay immediate dividends in speed without costing you significantly in terms of accuracy. As you continue to practice, you will find that your speed increases even more--but the focus should never be on TRYING to go fast, but rather on trying to be efficient.