The distribution of diffuse gas in the intergalactic medium (IGM) imprints a series of hydrogen absorption lines on the spectra of distant background quasars known as the Lyman-α forest. Cosmological hydrodynamical simulations predict that IGM density fluctuations are suppressed below a characteristic scale where thermal pressure balances gravity. We measured this pressure-smoothing scale by quantifying absorption correlations in a sample of close quasar pairs. We compared our measurements to hydrodynamical simulations, where pressure smoothing is determined by the integrated thermal history of the IGM. Our findings are consistent with standard models for photoionization heating by the ultraviolet radiation backgrounds that reionized the universe.