It doesn't make sense that your conditioning would have anything to do with it if you're already doing workouts that are more aerobic than cycling at a moderate pace.

The one thing that occurs to me is that your workouts are indoors. What's the weather like where you are? Is it possible you have asthma triggered by cold, humidity, pollen, car exhaust?

Fainting isn't good, and especially not on two wheels in traffic. Maybe get it checked out with your doctor?