Braking hard involves a massive transfer of weight to the front of the bike. So, in preparation, you move your body back as much as you can (not much on a sport bike) and brace yourself with the arms, right?
Wrong !! if you brace all your weight with the arms, it will mess with your control of the bike. Say you need to steer towards the end of the braking, to avoid something... as you're pulling 1g of deceleration, you won't be able to fight the handlebars and turn the bike.
You need to decouple yourself from the handlebars and brace your body in some other way. Grip the bike with the knees very tight and take some weight that way. Use your chest, if you can, like those tall tanks on super-sport bikes.
Note that it is generally a mistake to steer hard while braking hard - the loads on the front tire may cause it to loose traction - this is one of the big reasons for bike crashes, especially on the track. However, having seen many of those youtube videos whe bikers end up running into the car that cut them off, hard on the brakes, I think it's certainly worth trying, when push comes to shove!
So, first thing I do when something doesn't feel right? Grip the bike tight with my knees! That's good not not to brace for deceleration, but also, in general, to decouple your upper body from the bike and let the bike do its thing.