Thallium bromide (TlBr) semiconductor detectors are being developed as promising candidates for high-detection-efficiency, high-energy-resolution, and room-temperature gamma-ray spectrometers. This study presents methods for evaluating TlBr crystal quality and carrier transport characteristics using neutron Bragg-dip imaging and the time-of-flight method for pulsed-laser-induced carriers, respectively. Neutron Bragg-dip imaging effectively determines the crystal orientation distribution, revealing crystal imperfections and grain boundaries. Time-of-flight measurements provide a spatial distribution of carrier mobility. In this study, two samples obtained from both the upstream and downstream region in the crystal ingot were evaluated. Although both samples show similar crystal quality, the upstream sample showed high carrier mobility across all areas, whereas the downstream sample exhibits low mobility in some areas. These findings suggest that, at least within the range of carrier mobility currently obtained, the effect of crystal integrity on carrier mobility is less significant than that of impurities. In conclusion, combining neutron Bragg-dip imaging with carrier mobility measurements offers a comprehensive approach to evaluating and improving TlBr detectors.