As stated above, situational awareness is a big thing. I, as being one who's computer gaming roots go back to Aces of the Pacific (an old DOS WWII flight sim) and other Flight Sim games, the more you fly, the more you start to develop an inner eye for where your enemy is at in relation to your own craft. It may take a bit, but you will develop a sense of "Where they went and Where they are going'...once you can see it in your head, then all that is left to do is get your hands to coordinate with it.
Also, as others have also said, Flight Assist (FA) Off/On, will help you get better at swinging the nose around to get the pointy bits aimed at the bad guy faster. There are videos online made by Commander Moxen Wolf who is a Jedi Master at FA Off flying. He takes you through several videos to teach Commanders how to fly around in places I wouldn't try WITH FA On...Dude is GUD.
Best of luck Commander on your future battles, and here's to solid shields and weapons that never run dry!
Take care and check six Commanders o7
Michael Charlie Golf, signing off