Many posts with good suggestions, especially torso twisting. You also need to have situational awareness. Torso twisting is good only if the one shooting was in your frontal view (independent of the direction you were going). Twisting your torso so it lines it up for someone else shooting at you hurts too
Directional movement/positioning. If at all possible do not head directly towards the enemy. Learn to keep your range if you are primarily long range weapons, since walking up equipped with 2 gauss rifles and a few medium vs another mech with medium lasers/srm6/ac20 makes life really short. Also learn to use cover, both while moving forward or even fading back into cover.
Tics (weapon groupings) When feasible, it is a good idea to have tics to separate left and right side weapons, since this allows you to use weapons while taking cover.
Targets. Stick with one target as much as possible. This does not mean to pick a mech and single it out while the rest of your unit is targeting 2 other mechs. Switching and firing at different targets allows the enemy to live longer, giving them time to take down their own target, reducing your team's force multiplier. There are exceptions to be made especially if you have the drop on 1 or 2 snipers and you have some cover yourself. Grabbing their attention while avoiding taking too much damage means less damage to your other team mates, allowing them to live longer.