The galaxy cluster RX J0603.3+4214 at z=0.225 is one of the rarest clusters boasting an extremely large (~2 Mpc) radio-relic. Because of the remarkable morphology of the relic, the cluster is nicknamed Toothbrush Cluster. Although the clusters underlying mass distribution is one of the critical pieces of information needed to reconstruct the merger scenario responsible for the puzzling radio-relic morphology, its proximity to the Galactic plane b~10 deg has imposed significant observational challenges. We present a high-resolution weak-lensing study of the cluster with Subaru/Suprime Cam and Hubble Space Telescope imaging data. Our mass reconstruction reveals that the cluster is comprised of complicated dark matter substructures closely tracing the galaxy distribution, however in contrast with the relatively simple binary X-ray morphology. Nevertheless, we find that the cluster mass is still dominated by the two most massive clumps aligned north-south with a ~3:1 mass ratio (M_{200}=6.29_{-1.62}^{+2.24} x 10^{14} Msun and 1.98_{-0.74}^{+1.24} x 10^{14} Msun for the northern and southern clumps, respectively). The southern mass peak is ~2 offset toward the south with respect to the corresponding X-ray peak, which has a bullet-like morphology pointing south. Comparison of the current weak-lensing result with the X-ray, galaxy, and radio-relic suggests that perhaps the dominant mechanism responsible for the observed relic may be a high-speed collision of the two most massive subclusters, although the peculiarity of the morphology necessitates involvement of additional sub-clusters. Careful numerical simulations should follow in order to obtain more complete understanding of the merger scenario utilizing all existing observations.