Web testing has long been recognized as a notoriously difficult task. Even nowadays, web testing still heavily relies on manual efforts while automated web testing is far from achieving human-level performance. Key challenges in web testing include dynamic content update and deep bugs hiding under complicated user interactions and specific input values, which can only be triggered by certain action sequences in the huge search space. In this paper, we propose WebExplor, an automatic end-to-end web testing framework, to achieve an adaptive exploration of web applications. WebExplor adopts curiosity-driven reinforcement learning to generate high-quality action sequences (test cases) satisfying temporal logical relations. Besides, WebExplor incrementally builds an automaton during the online testing process, which provides high-level guidance to further improve the testing efficiency. We have conducted comprehensive evaluations of WebExplor on six real-world projects, a commercial SaaS web application, and performed an in-the-wild study of the top 50 web applications in the world. The results demonstrate that in most cases WebExplor can achieve a significantly higher failure detection rate, code coverage, and efficiency than existing state-of-the-art web testing techniques. WebExplor also detected 12 previously unknown failures in the commercial web application, which have been confirmed and fixed by the developers. Furthermore, our in-the-wild study further uncovered 3,466 exceptions and errors.