Fuzzing is an effective bug-finding technique but it struggles with complex systems like JavaScript engines that demand precise grammatical input. Recently, researchers have adopted language models for context-aware mutation in fuzzing to address this problem. However, existing techniques are limited in utilizing coverage guidance for fuzzing, which is rather performed in a black-box manner. This paper presents a novel technique called CovRL (Coverage-guided Reinforcement Learning) that combines Large Language Models (LLMs) with reinforcement learning from coverage feedback. Our fuzzer, CovRL-Fuzz, integrates coverage feedback directly into the LLM by leveraging the Term Frequency-Inverse Document Frequency (TF-IDF) method to construct a weighted coverage map. This map is key in calculating the fuzzing reward, which is then applied to the LLM-based mutator through reinforcement learning. CovRL-Fuzz, through this approach, enables the generation of test cases that are more likely to discover new coverage areas, thus improving vulnerability detection while minimizing syntax and semantic errors, all without needing extra post-processing. Our evaluation results indicate that CovRL-Fuzz outperforms the state-of-the-art fuzzers in terms of code coverage and bug-finding capabilities: CovRL-Fuzz identified 48 real-world security-related bugs in the latest JavaScript engines, including 39 previously unknown vulnerabilities and 11 CVEs.