We consider homogeneous Lorentz spaces of dimension at least 3. We prove that if such a space has ‘big’ isotropy (that is, a non-precompact and irreducible isotropy group), then this space must have constant sectional curvature. As a corollary, we obtain a new direct proof of the fact that irreducible Lorentz symmetric spaces have constant curvature, which was known via (algebraic) classification.