I think Sheldon hit it on the head with eliminating waste. You’ll find the top shooters don’t actually appear to be moving quickly, but they never time out. It’s about fluidity and economy of motion.
As an example, take the PRS barricade stage for instance since speed is particularly important here. Try this: get into your final shooting position where you are rock solid and feel confident you would make a hit...ok now work back from there. How do you get into that position most economically when approaching the barricade from 10 yards back (as per rules). When you get up to the barricade your left knee goes to the ground, your right knee stays up and your right elbow goes directly onto your right knee. This all happens in one fluid motion. You can practice that without even having a rifle in your hand. When you drop your rifle onto the barricade, it should already be on the magnification power with which you plan to shoot the stage. Now when you get onto the glass, you should see the target in your reticle. This is another time killer. So many new guys are in their scope searching for the target. You want to pick a bush/berm/tree etc that you can see with your naked eye that is near the target. Doing this allows you to know to a pretty good degree where to point your rifle. When you mount the barricade. As soon as you see the target in your reticle you should be closing your bolt. Don’t wait until you put your cross hair in the middle of the target. Also, when you close the bolt, do it smoothly enough that you don’t lose your sight picture. Break the shot but keep your cheek on the rifle, run your bolt smoothly again and send the second round. You can see there’s a lot that goes into just the first shot or two, but much of it translates to the next shot and the next.
If you’re saying your slow, and I had to guess, I’d say you might be shooting with too high magnification. Something MOST of us were guilty of when we started out.