We report on a class of quantum spin Hall insulators (QSHIs) in strained-layer InAs/GaInSb quantum wells, in which the bulk gaps are enhanced up to fivefold as compared to the binary InAs/GaSb QSHI. Remarkably, with consequently increasing edge velocity, the edge conductance at zero and applied magnetic fields manifests time reversal symmetry-protected properties consistent with the Z_{2} topological insulator. The InAs/GaInSb bilayers offer a much sought-after platform for future studies and applications of the QSHI.