Why do pilots typically start out in piston-engined aircraft, rather than jets or turboprops?
In a word, money.  Have you priced time in a turboprop or light jet, vs time in a C152 or other piston trainer?  Or actually buying one?  You can pick up a decent used single for about the cost of a new car, while turboprop or jet is likely to start at half a million, and go up from there.  And add maintenance costs to that.

Then there are a lot of pilots (I'm one) whose flying just doesn't include what a jet or turboprop is likely to be good at.  I like to fly into short strips, often grass or dirt (and sometimes no strip at all e.g. dry lake beds).  Is your jet or turboprop well suited for that?

Even for someone looking towards a career with the airlines, you can build a lot of the necessary time a lot cheaper in a single-engined prop plane.  For instance, one of my flight instructors started doing that, getting paid plus lots of experience dealing with the mistakes of student pilots.  Then he moved to a job flying a small cargo plane doing pickups for FedEx or someone like that, before getting enough hours to look at an airline job.