Atmospheric turbulence causes the majority of weather-related aircraft accidents. Climate models project large increases in clear-air turbulence as the jet streams become more sheared in response to climate change. However, climate models have coarser resolutions than the numerical weather prediction models that are used to forecast clear-air turbulence operationally, raising questions about their suitability for this purpose. Here we provide the first rigorous demonstration that climate models are capable of successfully diagnosing clear-air turbulence and its response to climate change. We use an ensemble of seven clear-air turbulence diagnostics to compare 38 years of historic turbulence diagnosed from climate model simulations and high-resolution reanalysis data. We find that the differences in turbulence between the climate model and reanalysis data are much smaller than the spread between the diagnostics. When using a climate model to calculate the probabilities (and their temporal trends) of encountering clear-air turbulence of any strength, at any flight cruising level, and in any season, we find that most of the uncertainty stems from the turbulence diagnostics rather than the climate model. These results confirm the suitability of climate models for the task of producing future clear-air turbulence projections. The turbulence increases are generally larger when diagnosed from the reanalysis data than the climate model, suggesting that previous quantifications from climate models of the response of clear-air turbulence to climate change may be underestimates. Our results show that the key to reducing uncertainty in projections of future clear-air turbulence lies in improving the clear-air turbulence diagnostics rather than the climate models.