Traditional models of vehicle dynamics engineered from physical principles are usually simplified and assumed, resulting in the model cannot accurately reflect the actual dynamic characteristics of the vehicle under some working conditions, affecting the control accuracy and even safety. In view of this, inspired by the single-track model, this paper uses the data-driven methods to establish a new high-performance time-delay feedback neural network vehicle dynamics model. The feedback connection of a network can describe complex dynamics. The multi-time-step input of the state and control can include highly nonlinear and strong coupling characteristics of a vehicle. The test results of modeling accuracy show that the proposed model can achieve higher vehicle dynamics prediction accuracy than nonlinear vehicle model. Different from the traditional vehicle dynamics model, the proposed model has long-term memory cells, which can implicitly predict coefficient of friction and can be applied to different road conditions. Then, the trajectory tracking control algorithm is designed based on the proposed vehicle model. According to the steady-state steering assumption, the feedforward front wheel steering angle is calculated, and the steady-state sideslip angle is integrated into the steering feedback according to the reference path to realize the reference trajectory tracking control. Finally, Simulink/CarSim is used to conduct the simulation analysis under the double lane change conditions to evaluate the proposed control algorithm. The analysis results show that the control algorithm based on the proposed model can achieve an accurate tracking control effect of a vehicle at medium and high speeds, providing high-accuracy track tracking and good lateral stability of intelligent vehicles.