Indirect detection of X-rays using single-crystal scintillators is a common approach for high-resolution X-ray imaging. With the high X-ray flux available from synchrotron sources and recent advances in high-speed visible-light cameras, these measurements are increasingly used to obtain time-resolved images of dynamic phenomena. The X-ray flux on the scintillator must, in many cases, be limited to avoid thermal damage and failure of the scintillator, which in turn limits the obtainable light levels from the scintillator. In this study, a transient one-dimensional numerical simulation of the temperature and stresses within three common scintillator crystals (YAG, LuAG and LSO) used for high-speed X-ray imaging is presented. Various conditions of thermal loading and convective cooling are also presented.