Several private Web search solutions have been proposed to preserve the user privacy while querying search engines. However, most of these solutions are costly in term of processing , network overhead and latency as they mostly rely on cryptographic techniques and/or the generation of fake requests. Furthermore, all these solutions protect all queries similarly, ignoring whether the original request contains sensitive content (e.g., religious, political or sexual orientation) or not. Based on an analysis of a real dataset of Web search requests, we show that queries related to sensitive matters are in practice a minority. As a consequence, protecting all queries similarly results in poor performance as a large number of queries get overprotected. In this paper, we propose a request sensitivity assessment module that we use for improving the practicability of existing private web search solutions. We assess the sensitivity of a request in two phases: a semantic sensitivity analysis (based on the topic of the query) and a request linkability analysis (based on the similarity between the current query and the query history of the requester). Finally, the sensitivity assessment is used to adapt the level of protection of a given query according to its identified degree of sensitivity: the more sensitive a query is, the more protected it will be. Experiments with a real dataset show that our approach can improve the performance of state-of-the-arts private Web search solutions by reducing the number of queries overpro-tected, while ensuring a similar level of privacy to the users, making them more likely to be used in practice.