I finished the game on a Mac, so it should be doable. But, I was forced to use a joystick, because the keyboard and mouse controls just didn't feel responsive enough. I didn't see you mention if you were using one, but I recommend it as the ideal peripheral for this game. Also, there are some cheat codes you can use to give yourself invincibility, and you can use that to practice until you get a feel for the gameplay and its nuances.

You have to lead the targets (the projectiles are slow so you have to compensate), also a joystick is the best way to play: mouse and keyboard controls are only there as a backup option basically (either a Logitech 3D Pro or a Thrustmaster T-16000m should do, the latter is sometimes sold with a throttle but you won't need or be able to use it for this game it also has magnetic sensors which are less prone to wear and tear than the potentiometers of the logitech model, they are both MAC compatible, windows only allows some extra functionalities that aren't necessary).
The asteroids basically are generated in front of you if you stay on the same course for too long and are generated more frequently the fatser you go, so never go too fast (and for the love of God avoid the afterburner) and always make sure to make small momevements so your ship never points exactly the same way. Mines work the same way and yes, it's stupid and one of the things I hated about these games but it's not insurmountable and can be used to your advantage (a risky manouver when dogfighting in an asteroid field is using the afterburner when you have an anemy right in fron of you so the game will generate an asteroid that will hit the enemy fighter before getting to you).

There is a "simulator" in the bar that lets you practice dogfighting before starting the game proper, just click on the arcade cabinet on the left.

Before going on a mission try playing around with the simulator. Getting the hang of the controls of an old game isn't trivial, it takes some time. When I started playing the game, my goal was defeating a couple of waves of each enemy types before I start playing actual missions. I think all you need is practice.

And try to listen to the NPCs and the manual, because some of their advice is useful (some less so, though). For example they tell you the optimal speed for asteroid fields (I forgot, it's somewhere from 300 to 400, not sure), because if you're too fast, your turning radius is too big, but if you're too slow, the asteroids catch up with you. So, what I usually do if I have to dogfight in an asteroid fields is set the speed at a safe level, and use the afterburner if I need to dodge enemy attacks.

Also, a little trick for dogfights in asteroid fields: the asteroids spawn with a speed opposite to your speed. So if you're chasing an enemy fighter, stepping on the afterburner might spawn an asteroid right in front of it, at a high speed. If you're aiming exactly at the enemy plane, no newly spawned asteroids can hit you, because the enemy ship will shield you, and get damaged in the process.

I'm also using a joystick, because that's what I got used to playing the X-Wing games. Though I could play Privateer with a mouse, because the Wing Commander franchise has somewhat better mouse controls (you control the speed of turning, not the turning itself iwth the mouse).