Autonomous car racing raises fundamental robotics challenges such as planning minimum-time trajectories under uncertain dynamics and controlling the car at its friction limits. In this project, we consider the task of autonomous car racing in the top-selling car racing game Gran Turismo Sport. Gran Turismo Sport is known for its detailed physics simulation of various cars and tracks. Our approach makes use of maximum-entropy deep reinforcement learning and a new reward design to train a sensorimotor policy to complete a given race track as fast as possible. We evaluate our approach in three different time trial settings with different cars and tracks. Our results show that the obtained controllers not only beat the built-in non-player character of Gran Turismo Sport, but also outperform the fastest known times in a dataset of personal best lap times of over 50,000 human drivers.