The flight crew was advised by air traffic control that the wind was from 320 degrees at 13 knots with gusts to 16 knots, and was cleared for a touch-and-go landing on runway 35. After touchdown with neutral aileron and rudder control input, the student glanced to remove carburetor heat and retract the flaps but looked outside of the cockpit and noted the airplane was moving towards the right side of the runway and was not lined up. The CFI took control of the airplane and attempted to correct but the right main landing gear collided with a snow bank at the right edge of the runway.