With the proliferation of social media platforms, users are exposed to vast information, including posts containing misleading claims. However, the pervasive noise inherent in these posts presents a challenge in identifying precise and prominent claims that require verification. Extracting the core assertions from such posts is arduous and time-consuming. We introduce a novel task called Claim Normalization (aka ClaimNorm) that aims to decompose complex and noisy social media posts into more straightforward and understandable forms, termed normalized claims. We propose CACN, a pioneering approach that leverages chain-of-thought and claim check-worthiness estimation, mimicking human reasoning processes, to comprehend intricate claims. Moreover, we capitalize on large language models' powerful in-context learning abilities to provide guidance and improve the claim normalization process. To evaluate the effectiveness of our proposed model, we meticulously compile a comprehensive real-world dataset, CLAN, comprising more than 6k instances of social media posts alongside their respective normalized claims. Experimentation demonstrates that CACN outperforms several baselines across various evaluation measures. A rigorous error analysis validates CACN's capabilities and pitfalls.