By regularly querying Web search engines, users (unconsciously) disclose large amounts of their personal data as part of their search queries, among which some might reveal sensitive information (e.g. health issues, sexual, political or religious preferences). Several solutions exist to allow users querying search engines while improving privacy protection. However, these solutions suffer from a number of limitations: some are subject to user re-identification attacks, while others lack scalability or are unable to provide accurate results. This paper presents CYCLOSA, a secure, scalable and accurate private Web search solution. CYCLOSA improves security by relying on trusted execution environments (TEEs) as provided by Intel SGX. Further, CYCLOSA proposes a novel adaptive privacy protection solution that reduces the risk of user re-identification. CYCLOSA sends fake queries to the search engine and dynamically adapts their count according to the sensitivity of the user query. In addition, CYCLOSA meets scalability as it is fully decentralized, spreading the load for distributing fake queries among other nodes. Finally, CYCLOSA achieves accuracy of Web search as it handles the real query and the fake queries separately, in contrast to other existing solutions that mix fake and real query results.