First-principles calculations are employed to investigate structural, electronic and topological insulating properties of XBi (X = B, Al, Ga, and In) monolayers upon halogenation. It is known that Y-XBi (X = Ga, In, Tl; Y = F, Cl, Br, I) can originate inversion-asymmetric topological insulators with large bulk band gaps. Our results suggest that Y-XBi (X = B, Al; Y = F, Cl, Br, I) may also result in nontrivial topological insulating phases. Despite the lower atomic number of B and Al, the spin-orbit coupling opens a band gap of about 400 meV in Y-XBi (X = B, Al), exhibiting an unusual electronic behavior for practical applications in spintronics. The nature of the bulk band gap and Dirac-cone edge states in their nanoribbons depends on the group-III elements and Y chemical species. They lead to a chemical tunability, giving rise to distinct band inversion symmetries and exhibiting Rashba-type spin splitting in the valence band of these systems. These findings indicate that a large family of Y-XBi sheets can exhibit nontrivial topological characteristics, by a proper tuning, and open a new possibility for viable applications at room temperature.