The new concept of “human-machine shared control” provides an amazing thinking to enhance driving safety, which has been attracted a great deal of research effort in recent years. However, little attention has been paid to the nonlinearity of the shared control system brought by the tire, which significantly influences the control performance under extreme driving conditions. This paper presents a novel shared steering torque control scheme to model the human-machine steering torque interaction near the vehicle’s handling limit, where both driver and driver assistance system (DAS) are exerting steering torque to maneuver the vehicle. A six-order driver-vehicle dynamic system is presented to elaborate the relationship between steering torque input and vehicle lateral motion response. Particularly, we use a piecewise affine (PWA) method to approximate the tire nonlinearity. Based on switched model predictive control (sMPC) method, the proposed hybrid PWA system is taken to model the shared control problem as a leader-follower game to minimize the driver-DAS steering torque conflicts. On this basis, the constrained Stackelberg equilibrium solution of the shared steering torque control problem is derived to describe the path-tracking behaviors of driver and DAS. The co-simulation using Simulink and CarSim is conducted to validate the proposed shared steering torque control strategy. While ensuring the acceptable tracking performance, the proposed shared control scheme effectively stabilizes the vehicle under emergency path tracking maneuver. The shared controller in this paper can effectively compensate the weakness of the linear shared controller in dealing with tire nonlinearity and is conducive to the design of the steering system in the future co-driving intelligent vehicle.