Autonomous racing has gained increasing attention in recent years, as a safe environment to accelerate the development of motion planning and control methods for autonomous driving. Deep learning models, predominantly based on neural networks (NNs), have demonstrated significant potential in modeling the vehicle dynamics and in performing various tasks in autonomous driving. However, their black-box nature is critical in the context of autonomous racing, where safety and robustness demand a thorough understanding of the decision-making algorithms. To address this challenge, this paper proposes MS-NN-steer, a new Model-Structured Neural Network for vehicle steering control, integrating the prior knowledge of the nonlinear vehicle dynamics into the neural architecture. The proposed controller is validated using real-world data from the Abu Dhabi Autonomous Racing League (A2RL) competition, with full-scale autonomous race cars. In comparison with general-purpose NNs, MS-NN-steer is shown to achieve better accuracy and generalization with small training datasets, while being less sensitive to the weights' initialization. Also, MS-NN-steer outperforms the steering controller used by the A2RL winning team. Our implementation is available open-source in a GitHub repository.