Having driven a CX-50 with the 20's and another with the 17's, and CX-5's of both generations with the 19's and 17's, AND having swapped my 2nd Gen CX-5 19's for 20's with a completely different (and better) tire, I can tell you with 100% certainty that what you are describing is the fault of the 225/55/19 Toyo A36 on the 19x7" wheels.
I wasn't happy with the feeling that the rear of the car and the front seemed to respond at different rates, with the back lagging behind the front, when shifting direction at highway speeds. The main problem is that it's not a sporty tire and responds slowly.
My lowering springs with higher rates probably helped with this as well, so consider that into the equation. The car is much tighter now and responds instantly, both ends in sync.
So combine soft springs and soft sidewalls and that's what you will get. Get in a CX-50 with the 20's and it feels completely different. That car is more direct, with very little body roll, yet a bit harder ride quality than the CX-5, partly due to losing the IRS and partly due to the wheels. It feels like the sway bars may be stiffer, too, because I don't know how else to account for the lack of body roll. This provided a more enjoyable turning experience, in my opinion.