In this work, we propose a self-improving artificial intelligence system to enhance the safety performance of reinforcement learning (RL)-based autonomous driving (AD) agents using black-box verification methods. RL algorithms have become popular in AD applications in recent years. However, the performance of existing RL algorithms heavily depends on the diversity of training scenarios. A lack of safety-critical scenarios during the training phase could result in poor generalization performance in real-world driving applications. We propose a novel framework in which the weaknesses of the training set are explored through black-box verification methods. After discovering AD failure scenarios, the RL agent's training is re-initiated via transfer learning to improve the performance of previously unsafe scenarios. Simulation results demonstrate that our approach efficiently discovers safety failures of action decisions in RL-based adaptive cruise control (ACC) applications and significantly reduces the number of vehicle collisions through iterative applications of our method. The source code is publicly available at https://github.com/data-and-decision-lab/self-improving-RL.