First, I doubt the CVT used in our cars will handle any significant increase in horsepower. If you understand how CVTs operate, you'll understand why. CVTs don't have actual gears and are only built for fuel economy. The only exception is the CVT used in the new Corolla HB which has a physical 1st gear.
Second, it's not simple to just throw a turbo onto any engine. You have to make sure the engine internals are built to handle forced induction. Many times, the compression ratio of a forced induction engine is lowered compared to a naturally aspirated one. Also, you have to see if any of the cast parts such as the piston, connecting rod, and crank are strong enough to deal with the increased stresses from forced induction. I highly doubt there are any forged internals used in this motor.
Third, fueling and mapping. Unless you have access to a dyno or find some tuner which will spend the time to create a proper map for this engine using forced induction, you're wasting your time.
I don't want to sound callous, but if you wanted a performance car, you should have purchased one from the beginning. If you look at my list of gripes about this car in another thread, I bring up the poor choice of engine by Toyota for this platform. Especially since the chassis is so good handling wise and will support tons of power as evidenced by a tuner who converted a C-HR into rear wheel drive and threw in a turbo 4 producing 600 HP. If you have any notion of trying this, you better have a big bank account. And after all the money that is spent, you'd be further ahead by buying a car with the performance from the factory as stock. I've done tons of modding in my past. I learned a lot about cars doing this. I modd'ed a 94 Z24 with the 60 degree V6. 1.6 ratio Crane roller rockers, Crane Compucam 2030, attempted all different kinds of performance chips, and 50 shot of dry spray nitrous. While it was kind of fun building an absolute sleeper, as I got older I recognize how much money I wasted in building up that car. My next cars were a 94 Z28 with the 350 small block Chevy LT1 motor and now with my 13 BMW 135i. My 19 C-HR's engine remains unmolested and will stay so as it's not worth the aggravation or the possible reduction in reliability. If someone does a flash tune or some sort of piggy back tune which I can see an actual dyno chart on along with a few real world testimonials, I would consider it. Tearing into the motor? No thanks.
2019 Toyota C-HR LE Silver Knockout Metallic
2013 BMW 135i Carbon Black
2009 Ducati 848 Pearl White/Black
2004 Kawasaki Ninja ZX-10R Black