Trajectory data has the potential to greatly benefit a wide-range of real-world applications, such as tracking the spread of the disease through people's movement patterns and providing personalized location-based services based on travel preference. However, privay concerns and data protection regulations have limited the extent to which this data is shared and utilized. To overcome this challenge, local differential privacy provides a solution by allowing people to share a perturbed version of their data, ensuring privacy as only the data owners have access to the original information. Despite its potential, existing point-based perturbation mechanisms are not suitable for real-world scenarios due to poor utility, dependence on external knowledge, high computational overhead, and vulnerability to attacks. To address these limitations, we introduce LDPTrace, a novel locally differentially private trajectory synthesis framework. Our framework takes into account three crucial patterns inferred from users' trajectories in the local setting, allowing us to synthesize trajectories that closely resemble real ones with minimal computational cost. Additionally, we present a new method for selecting a proper grid granularity without compromising privacy. Our extensive experiments using real-world data, various utility metrics and attacks, demonstrate the efficacy and efficiency of LDPTrace.