The proliferation of GPS-enabled devices and advances in positioning technologies have greatly facilitated the collection of user location data, making them valuable across various domains. One of the most common and practical uses of these location datasets is to recommend the most probable route between two locations to users. Traditional algorithms for route recommendation rely on true trajectory data collected from users, which raises significant privacy concerns due to the personal information often contained in location data. Therefore, in this paper, we propose a novel framework for computing optimal routes using location data collected through differential privacy (DP)-based privacy-preserving methods. The proposed framework introduces a method for accurately extracting transitional probabilities from perturbed trajectory datasets, addressing the challenge of low data utility caused by DP-based methods. Specifically, to effectively compute transitional probabilities, we present a density-adjusted sampling method that enables the collection of representative data across all areas. In addition, we introduce an effective scheme to approximately estimate transitional probabilities based on sampled datasets. Experimental results on real-world data demonstrate the practical applicability and effectiveness of our framework in computing optimal routes while preserving user privacy.