Climate-model simulations exhibit approximately two times more tropical tropospheric warming than satellite observations since 1979. The causes of this difference are not fully understood and are poorly quantified. Here, we apply machine learning to relate the patterns of surface-temperature change to the forced and unforced components of tropical tropospheric warming. This approach allows us to disentangle the forced and unforced change in the model-simulated temperature of the midtroposphere (TMT). In applying the climate-model-trained machine-learning framework to observations, we estimate that external forcing has produced a tropical TMT trend of 0.25±0.08 K decade^-1 between 1979 and 2014, but internal variability has offset this warming by 0.07±0.07 K decade^-1. Using the Community Earth System Model version 2 (CESM2) large ensemble, we also find that a discontinuity in the variability of prescribed biomass-burning aerosol emissions artificially enhances simulated tropical TMT change by 0.04 K decade^-1. The magnitude of this aerosol-forcing bias will vary across climate models, but since the latest generation of climate models all use the same emissions dataset, the bias may systematically enhance climate-model trends over the satellite era. Our results indicate that internal variability and forcing uncertainties largely explain differences in satellite-versus-model warming and are important considerations when evaluating climate models.