Why does flight training usually start with (unsafe?) propeller planes and not with (safer?) jets?
Light Propeller Aircraft, if flown within their design limits, are just as safe as jets. If you do a proper and thorough daily inspection, you are far less likely to suffer any form of mechanical failure in the air. In saying this, the majority of all crashes, be it in a Jet Powered Airliner or a Propeller Driven Light Aircraft are caused due to pilot error.

You start out learning in a light single as:


They are lighter, and slower and can usually take the beating handed out by inexperienced pilots, this being:


Hard landings
Slipping onto the runway
Just hammering the day lights out of the engine.

Jet Aircraft are extremely expensive, and generally fairly heavy. Inexperienced pilots generally pay AU$80,000 for a Commercial Pilot License (In Australia) which qualifies you to fly single engine aircraft not over 5,700kg under the VFR by day. This is an absolute, level entry qualification looking for a job (most people will find that they at least also need a night VFR rating or a MECIR), so to add the cost of doing training in a jet on top of that, even if it is just for PPL, will be massive.
Jet Aircraft will be insured to be flown by pilots who have an amount of total experience and experience on aircraft/engine type. So, again, unless you're willing to pay the massive excess and premiums to be put on the insurance, you're going to have to fly a prop.


Now, you have identified that a lot of crashes are light singles (i.e. Cessna). This is because generally, these aircraft are not flown with the same discipline as a Jet Aircraft. Also, the pilots flying them are generally lower hour pilots who do not have as much experience (again, this is a generalisation, not an exact figure of the total community).

As for starting in a Jet, I honestly doubt that there is any flight school on the planet that will allow you to start training in a Jet (Even Fighter Jet pilots start out flying tiny little single engine aircraft).