Further to Steve's answer...

It is also worth analysing what it is, exactly, that you are reacting to. Novice fighters often concentrate on the opponent's fists or feet, whereas more experienced fighters tend to have a more 'total body' awareness.

I'll use boxing as an example here. If I concentrate on the fists, it is easy to get caught up in a dangerous process of prediction, in which you try to guess which hand is going to launch at you. To guess reliably is of course impossible.

Tremendous progress can be gained simply from changing your point of focus from the limbs to the sternum (roughly, the centre of the chest). When you do this, you begin to identify the ways in which the body telegraphs strikes from the hips and shoulders. It may not seem like much, but the consequences for your ability to react effectively are enormous.

Another way to gain reaction time is of course to maintain non-predictable (erratic) movement patterns, so that your opponent's strikes are hampered my a constant need to adjust to your location.

One of the best ways of all is to simply stay out of range. This is not really possible for a purely offensive fighter, but it is for a counter-puncher or a defensive fighter.

Lastly, learning to relax in the face of aggression will transform your defensive ability. When you are tense, you slow down a lot. Your ability to evade is almost crippled. I'm not saying you should be able to shed all fear (which can be very useful), but that you learn to defend without 'seizing' up.  This requires an experienced and disciplined sparring partner, one who can challenge you without trying to 'win' or hurt you.