I think you just lack an understanding of turn-dynamics. Consider this, all AH pilots black out at the same 6G's, they all also experience tunnel vision at 5G's+. So, if you are blacked out and the other guy isn't then you are turning "harder" than he is. Each aircraft in AH can pull 6G's, it just depends on the speed they require to do so in order to generate the lift required. Some aircraft that are lighter, or generate lift better, can pull 6G's at lower speeds than others. Generally though, at 250mph+, every fighter can pull 6G's, at least for a brief time.
So, how does the other person stay with you and you can't with them, well, that has to do with pursuit angles. There are 3 basic pursuits available, a lead, a pure, and a lag. If you think about this in a race-car/corner concept, the lead is going for the inside lane, the pure is following, and the lag is the high-line. You sound like you pull lead pursuit but the impact of that is that you have to turn harder to maintain that angle, thus you are pulling more G's and burning more energy generating lift. That's why you black out first and never get your guns to bear. Add to that a situation where the attacker generally has more speed and he has more potential to generate lift and is more likely to blackout (over-generate). A pure pursuit is where you follow the guy, you pull no more/less than he does, thus you both would be blackout out at the same time. Of course, this doesn't allow you a shot because you aren't aiming "infront" of the enemy... that said, you can use this to transition later to the lead pursuit so you can take your shot. Lots of people use this as a "settling" phase, they let the enemy struggle around infront while they settle in and wait. Once they see their opportunity, they pull Lead Pursuit for just the amount of time they need to take their shot. A Lag Pursuit is even less often understood, this is the "high-line" where you pull a larger radius but still retain a high turn-rate. This is very useful for controlling closure rates so you don't "pass" the enemy accidentally. You can carry higher speeds in a lag, thus retaining more energy, and then when you are ready you pull Lead Pursuit and take your shot.
I would say, by your description, that you attempt to pull Lead too much and thus black yourself out going for a quick kill at a difficult angle/closure rate. Good pilots will exploit this in their defense knowing you "can't" make that shot (nobody could, everyone would black out just like you do). The people you encounter on the other side of the coin, who can follow you, likely are using an less aggressive turn (pure or lag) and wait for you to slow a bit, then then pull lead angle and shoot you. they use some understanding of pursuit angles and patiences to let you squirm and play out your initial defense, then they jump on you.
A side note, Pursuit type also impacts closure rates so is very useful for controlling how quickly you are catching an enemy in a turn. If you Lead Pursuit, you are cutting the corner (shorter path) and thus your closure rate increases. Pure Pursuit would maintain your closure rate (based on speed differential) and a Lag Pursuit would actually slow your closure (maybe even allowing the distance to increase because your flight path is longer than the guy you are chasing).
If you want a demo, drop me a note and I can meet you in the Training Arena and show you. It's a very important concept but once you see it in action it usually all makes sense. Feel free to email me any questions if you want.
soda@hitechcreations.com