A quantum spin Hall (QSH) insulator with high stability, large bulk band gap and tunable topological properties is crucial for both fundamental research and practical application due to the presence of dissipationless edge conducting channels. Recently, chemical functionalization has been proposed as an effective route to realize the QSH effect. Based on first-principles calculations, we predict that a two-dimensional TlP monolayer would convert into a topological insulator with the effect of bromination, accompanied by a large bulk band gap of 76.5 meV, which meets the requirement for room-temperature application. The topological nature is verified by the calculation of Z2 topological invariant and helical edge states. Meanwhile, an appreciable Rashba spin splitting of 77.2 meV can be observed. The bulk band gap can be effectively tuned with external strain and electric field, while the Rashba spin splitting shows a parabolic variation trend under an external electric field. We find that the topological property is available for the TlP film when the coverage rate is more than 0.75. BN and SiC are demonstrated as promising substrates to support the topological nature of TlPBr2 film. Our findings suggest that a TlPBr2 monolayer is an appropriate candidate for hosting the nontrivial topological state and controllable Rashba spin splitting, and shows great potential applications in spintronics.