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In a weighted logrank test, such as the Harrington-Fleming test and the Tarone-Ware test, predetermined weights are used to emphasize early, middle, or late differences in survival distributions to maximize the test's power. The optimal weight function under an alternative, which depends on the true hazard functions of the groups being compared, has been derived. However, that optimal weight function cannot be directly used to construct an optimal test since the resulting test does not properly control the type I error rate. We further show that the power of a weighted logrank test with proper type I error control has an upper bound that cannot be achieved. Based on the theory, we propose a weighted logrank test that self-adaptively determines an "optimal" weight function. The new test is more powerful than existing standard and weighted logrank tests while maintaining proper type I error rates by tuning a parameter. We demonstrate through extensive simulation studies that the proposed test is both powerful and highly robust in a wide range of scenarios. The method is illustrated with data from several clinical trials in lung cancer.