The exploitation of user search queries by search engines is at the heart of their economic model. As consequence, offering private Web search functionalities is essential to the users who care about their privacy. Nowadays, there exists no satisfactory approach to enable users to access search engines in a privacy-preserving way. Existing solutions are either too costly due to the heavy use of cryptographic mechanisms (e.g., private information retrieval protocols), subject to attacks (e.g., Tor, TrackMeNot, GooPIR) or rely on weak adversarial models (e.g., PEAS). This paper introduces X-Search, a novel private Web search mechanism building on the disruptive Software Guard Extensions (SGX) proposed by Intel. We compare X-Search to its closest competitors, Tor and PEAS, using a dataset of real web search queries. Our evaluation shows that: (1) X-Search offers stronger privacy guarantees than its competitors as it operates under a stronger adversarial model; (2) it better resists state-of-the-art re-identification attacks; and (3) from the performance perspective, X-Search outperforms its competitors both in terms of latency and throughput by orders of magnitude.