In the aftermath of an earthquake, one of the most critical needs is medical care for the injured. A large number of individuals require immediate attention, often overwhelming the available healthcare resources. The sudden surge in demand, coupled with limited resources, route congestion and infrastructure damage, makes immediate medical care provision a significant challenge. This challenge intensifies when fairness concerns exist. Temporary hospitals help address the need for medical care when existing healthcare infrastructure is inadequate and road access is restricted due to congestion and damage. We develop a mathematical program to locate temporary hospitals, determine transportation routes, and allocate the injured efficiently and equitably. The impact of route congestion on transportation times is modeled using the Bureau of Public Roads (BPR) function. An exact approach based on Benders decomposition is employed to solve the resulting mixed-integer second-order cone program, with various acceleration techniques to enhance the method. This methodology is applied to test problems and the anticipated Istanbul Earthquake case study, focusing on aspects such as hospital capacities, candidate locations, road network disruptions, and varying injury severity levels. The goal is to construct a comprehensive healthcare response plan. Numerical results demonstrate the importance of a holistic approach to ensure low costs, manage congestion, and equitable provision of healthcare.