When quantifying the importance of supply and demand for oil price fluctuations, a wide range of estimates have been reported. Models identified via a sharp upper bound on the short‐run price elasticity of supply find supply shocks to be minor drivers. In turn, when replacing the upper bound with a weakly informative prior, supply shocks turn out to be substantially more important. In this paper, I revisit the evidence in a model that combines weakly informative priors with identification by non‐Gaussianity. For this purpose, a SVAR is developed where the unknown distributions of the structural shocks are modeled nonparametrically. The empirical findings suggest that once identification by non‐Gaussianity is incorporated into the model, posterior mass of the short‐run oil supply elasticity shifts toward zero and oil supply shocks become minor drivers of oil prices. In terms of contributions to the forecast error variance of oil prices, the model arrives at median estimates of just 6% over a 16‐month horizon.