The only way to get faster is to go faster.
Grip the gun hard. All the top shooters have very strong grips which they apply to the guns they are shooting. They may not be using 100% of their grip strength, but even if they are only using 80% it's likely a lot more force than you are currently generating with 100% of your grip strength. So grip the gun hard.
The right grip helps the gun track predictably. In other words, the front sight lifts as the slide cycles and drops right back to where it was when you fired the last shot if your grip is good. If this isn't happening (and for many it isn't) your grip mechanics need work.
A significant level of grip strength applied to the grip allows you to aggressively work the trigger...so the sights aren't disrupted as you work through the trigger pull at speed.
There's also a mental aspect to this. You have to learn to "see" what you need to see at speed. In other words, in the fractions of a second that the sights are in motion to recognize when you have the sight picture you need to make the shot called for and then to execute a good enough trigger press to make that shot. Your brain's bandwidth for processing the visual and tactile information coming at you increases as you get more experience doing it.
In Aim Fast Hit Fast, one of the exercises Todd had his students do was a few mag dumps. The goal was to obtain a solid grip and then to fire the pistol literally as fast as they could make the gun work while pointed at the berm. The goal was to give the student an understanding of how fast they could make the gun go bang, and to have them focus visually on something on the gun...like trying to see the empty cases eject from the pistol. If you can make the gun work with sub .20 splits and see the shells streaming out of the side of the pistol, you can shoot it with reasonable accuracy at that speed, too. All you have to do is watch the sights.
To use Todd's favorite car analogy, driving smooth at 35 miles per hour does not prepare you to take a corner at 135 miles per hour. At some point to get faster you have to go faster. You have to familiarize your brain and body with the demands of operating at a higher level. If you ever watch Top Gear's episodes where they put a host in a Formula 1 car, they talked about their inability to "think" as fast as the car goes. They were, in other words, unaccustomed to making decisions and inputs at the speed necessary to pilot the vehicle at the speeds it was capable of going. They lacked confidence on when to make inputs and how much input to make.
Same thing applies to the gun. You have to learn to aim fast, work the trigger fast, recover fast, and repeat the cycle. A couple of weeks ago at ESS I used the NRA bullseye target at 10 yards to work on this. I worked on my grip and my trigger pull at speed and by the end of the session I was firing the gun at just a little bit under my absolute physical limit (meaning the physical limitation of how fast I can make the gun go bang doing nothing but moving my trigger finger as fast as possible) while keeping all the shots within the black. My splits ranged from .15-.17 which won't set records but is plenty damn fast by any measure...and those shots were aimed. They were aimed because I had learned how to see a "good enough" sight picture, to trust it, and to make a "good enough" trigger pull to keep those shots in the black.