The Mazda remote start does not have a turbo timer. I recently had an aftermarket remote start system pulled out and the Mazda one put in it's place (long story, but it works MUCH better with the smart-key system and they didn't have to hide one of my auxiliary keys in the car)... and I can confirm it doesn't have a turbo timer.
Here's the description from Mazda:
"Allows you to remotely start your vehicle, pre-heating or pre-cooling the vehicle's interior before you enter it. "Quick-Stop" feature allows you to keep your car running and locked while making a quick stop. "Car Find" feature flashes parking lights and beeps horn to help you find your car. Range is up to 800 feet."
I have no idea how to use the "Quick-Stop" feature. LOL