Using combined data from the Relativistic Heavy Ion and Large Hadron Colliders, we constrain the shear and bulk viscosities of quark-gluon plasma (QGP) at temperatures of ${sim,}150{-}350$ MeV. We use Bayesian inference to translate experimental and theoretical uncertainties into probabilistic constraints for the viscosities. With Bayesian Model Averaging we account for the irreducible model ambiguities in the transition from a fluid description of the QGP to hadronic transport in the final evolution stage, providing the most reliable phenomenological constraints to date on the QGP viscosities.