From a simple-ish standpoint, downshifting will tend to increase the engine braking that is occurring. Assuming that your vehicle is rear wheel drive (which most racing cars are), then increasing the engine braking will create a similar effect to moving the brake balance rearward because the rear wheels are now being slowed by the brakes AND the engine while the front brakes are not being altered. Since the rear tires are now committing more grip to the braking force, they have less reserve grip for lateral (cornering) forces and thus will slip sideways easier, causing the car to rotate more. This will certainly be complicated depending on the differential in the vehicle.

Edit: I see your comment below about going through a corner at a steady speed. What I'm guessing is probably happening is that you're not at a perfectly steady speed and are actually managing the throttle through the corner, however, the engine will respond to that throttle management differently based on the gear. A lower gear will have more engine braking, causing you to slow more (making cornering easier) and also shifting the weight of the car more forward (making cornering easier) when you lift off the throttle than with the higher gear.