Due to the increasing sophistication of web attacks, Web Application Firewalls (WAFs) have to be tested and updated regularly to resist the relentless flow of web attacks. In practice, using a brute-force attack to discover vulnerabilities is infeasible due to the wide variety of attack patterns. Thus, various black-box testing techniques have been proposed in the literature. However, these techniques suffer from low efficiency. This paper presents Reinforcement-Learning-Driven and Adaptive Testing (RAT), an automated black-box testing strategy to discover injection vulnerabilities in WAFs. In particular, we focus on SQL injection and Cross-site Scripting, which have been among the top ten vulnerabilities over the past decade. More specifically, RAT clusters similar attack samples together. It then utilizes a reinforcement learning technique combined with a novel adaptive search algorithm to discover almost all bypassing attack patterns efficiently. We compare RAT with three state-of-the-art methods considering their objectives. The experiments show that RAT performs 33.53% and 63.16% on average better than its counterparts in discovering the most possible bypassing payloads and reducing the number of attempts before finding the first bypassing payload when testing well-configured WAFs, respectively.