Traction, sidesliding
I'm guessing it's not the Kevlar belt that's causing the issue so much as the correlation between a Kevlar belt and an overall stiff tire. One of the most important qualities of a tire is its suppleness: how easily it deforms and contours itself to the uneven surface of the road. By deforming easily, you maximize surface area and increase grip. By adding lots of layers of puncture protection, some tires become rigid and don't handle sideways load all that well.

I don't necessarily recommend different tires: only you can determine whether the added puncture protection is worth taking more gradual turns. Push down hard on the outside pedal while cornering, and avoid taking turns over manhole covers, raised paint, train tracks, or any other type of road adornment, because they all can be pretty slick.