We report transport measurements of strained InAs/In$_{x}$Ga$_{1-x}$Sb composite quantum wells (CQWs) in the quantum spin Hall phase, focusing on the control of the energy gap through structural parameters and an external electric field. For highly strained CQWs with $x = 0.4$, we obtain a gap of 35 meV, an order of magnitude larger than that reported for binary InAs/GaSb CQWs. Using a dual-gate configuration, we demonstrate an electrical-field-driven topological phase transition, which manifests itself as a re-entrant behavior of the energy gap. The sizeable energy gap and high bulk resistivity obtained in both the topological and normal phases of a single device open the possibility of electrical switching of the edge transport.