Why do my low profile tires get flats much, much more than my other (regular) profile tires on other cars?
They Have A Smaller Volume Of Air.
CSA
They’re probably wider, so the cover more of the road, more chance to pick up a nail. Are all your flats of the nail/screw/etc variety? Or are some from potholes/curbs? Low profile tires are more easily damaged by those, too.
While I am not a tire expert…I will add…
Low profile tires are generally deigned with a comprise towards performance. They generally have a softer tread compound and may have less tread depth for less tread squirm under hard cornering.
Of course if you are driving hard on a car with performance tires, you are using all the road…you know turn-in, apex, and track-out line…might bring you to less used edges of the road where as David Hobbs says, “are full of “Clag””. Sorry whatching F1 on Speed right now. 
As you can see there are a number of possible reasons. I will add a few more. Since the distance between the tread the the wheel, there is more chance that hitting a pot hole will cause damage to the tyre, and/or the wheel as well. Also it appears that at least some of those tyres while more expensive are actually not as well constructed as standard tyres since the buyers are usually only looking for appearances.
Even low profile tires made by Michelin do not last as long due to the softer rubber. Agree that the low profile has less bounce to absorb shocks, and is more easily damaged.
In addition, people with low profile tires drive more vigorously and as a result may damage their tires more.
Softer compound of the tire for better road adhesion will cut easier, pick up more nails and stuff. Rocketman
How low a profile are we talking here?
What exactly have been the causes of the flats?
If they’re foreign object punctures you can’t very well blame the tires. If they’re pinch flats, perhaps you’ve gone to too low a profile for your driving environment…or need to slow down.
I’ve driven 45 series tire for four years now, in NH, and even with all our potholes the only flats I’ve had were from nails and screws when we had a construction project ongoing at work.
I don’t believe that low profile tires get more flats. That has not been my experience.
I’d avoid anything lower than 60 profile on an “ordinary” car. My Matrix has 55 profile tires and I wish there were a straightforward way I could go to 60 or 65. I remember my 81 Accord I went from 80 to 70 profile plus Tokiko struts and heavier sway bars. It cornered like a go-cart. You don’t need those super low tires to get good handling.