Data-driven generation of molecules with desired properties, also known as inverse molecular design (IMD), has attracted significant attention in recent years. Despite the significant progress in the accuracy and diversity of solutions, existing IMD methods lag behind in terms of trustworthiness. The root issue is that the design process of these methods is increasingly more implicit and indirect, and this process is also isolated from the native forward process (NFP), the ground-truth function that models the molecular dynamics. Following this insight, we propose TrustMol, an IMD method built to be trustworthy. For this purpose, TrustMol relies on a set of technical novelties including a new variational autoencoder network. Moreover, we propose a latent-property pairs acquisition method to effectively navigate the complexities of molecular latent optimization, a process that seems intuitive yet challenging due to the high-frequency and discontinuous nature of molecule space. TrustMol also integrates uncertainty-awareness into molecular latent optimization. These lead to improvements in both explainability and reliability of the IMD process. We validate the trustworthiness of TrustMol through a wide range of experiments.