The increasing reliance on Large Language Models (LLMs) for health information seeking can pose severe risks due to the potential for misinformation and the complexity of these topics. This paper introduces KnowNet a visualization system that integrates LLMs with Knowledge Graphs (KG) to provide enhanced accuracy and structured exploration. Specifically, for enhanced accuracy, KnowNet extracts triples (e.g., entities and their relations) from LLM outputs and maps them into the validated information and supported evidence in external KGs. For structured exploration, KnowNet provides next-step recommendations based on the neighborhood of the currently explored entities in KGs, aiming to guide a comprehensive understanding without overlooking critical aspects. To enable reasoning with both the structured data in KGs and the unstructured outputs from LLMs, KnowNet conceptualizes the understanding of a subject as the gradual construction of graph visualization. A progressive graph visualization is introduced to monitor past inquiries, and bridge the current query with the exploration history and next-step recommendations. We demonstrate the effectiveness of our system via use cases and expert interviews.