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Effective query expansion for web search benefits from promoting both exploration and diversity to capture multiple interpretations and facets of a query. While recent LLM-based methods improved retrieval performance and demonstrate strong domain generalization ability without additional training, they often generate narrowly focused expansions that overlook these properties due to knowledge anchoring within the model. We propose ThinkQE, a test-time query expansion framework addressing this limitation through two key components: a thinking-based expansion process that encourages deeper and comprehensive semantic exploration, and an evolving interaction strategy that iteratively refines expansions using retrieval feedback from the corpus. Experiments on diverse web search benchmarks (DL19, DL20, and BRIGHT) show ThinkQE consistently outperforms prior approaches, including training-intensive dense retrievers and rerankers.