I typically ride on the hoods, but I've been working lately on a more aero position. When in the drops or using clip-on aero bars I find that my quads get fatigued much more than when I'm in a normal road bike position. Is this normal and I just need to train more or am I missing something?
You likely need a bike fit. The body position for using aero should be quite different than riding on hoods. Unfortunately, there really isn't a good compromise between a road and a TT/aero position.
I recommend getting a professional bike fit. I'm including the "quick fit" below, but there are too many factors in play and millimeters matter. If possible, go to a shop that specializes in setting up Triathlon athletes (road fit != TT fit).
The "quick fit" for aero is to slam your saddle forward and then adjust height and reach. Your forearms should be on the pads, not your elbows. You should be able to reach all controls (brakes/shifting) without having to move your body (arms yes, body no)...if you can't, then you likely can't reach controls fast enough to be safe.