Compactness of families of solutions --- or of approximate solutions --- is a feature that distinguishes certain classes of nonlinear hyperbolic equations from the case of linear hyperbolic equations, in space dimension one. This paper shows that some classical compactness results in the context of hyperbolic conservation laws, such as the Lax compactness theorem for the entropy solution semigroup associated with a nonlinear scalar conservation laws with convex flux, or the Tartar-DiPerna compensated compactness method, can be turned into quantitative compactness estimates --- in terms of epsilon-entropy, for instance --- or even nonlinear regularization estimates. This regularizing effect caused by the nonlinearity is discussed in detail on two examples: a) the case of a scalar conservation law with convex flux, and b) the case of isentropic gas dynamics, in space dimension one.