We introduce ClaimCheck, an LLM-guided automatic fact-checking system designed to verify real-world claims using live Web evidence and small language models. Unlike prior systems that rely on large, closed-source models and static knowledge stores, ClaimCheck employs a transparent, stepwise verification pipeline that mirrors human fact-checking workflows consisting of Web search query planning, Web-based evidence retrieval and summarization, evidence synthesis and re-retrieval, and claim verdict evaluation. Each module is optimized for small LLMs, allowing the system to deliver accurate and interpretable fact-checking with significantly lower computational requirements. Despite using a much smaller Qwen3-4B model, ClaimCheck achieves state-of-the-art accuracy of 76.4% on the AVeriTeC dataset, outperforming previous approaches using LLaMA3.1 70B and GPT-4o. Extensive ablations demonstrate that careful modular design and prompting strategies can overcome the limitations of smaller LLMs. To promote accessibility and transparency, we provide a public demo at https://idir.uta.edu/claimcheck.