We report improved measurements of the all flavor astrophysical neutrino spectrum with IceCube by combining complementary neutrino samples in two independent analyses. Both analyses show evidence of a harder spectrum at energies below ∼30 TeV compared to higher energies where the spectrum is well characterized by a power law. The spectrum is better described by a log parabola or a broken power law, the latter being the preferred model. Both, however, reject a single power law over an energy range 5 TeV-10 PeV with a significance >4σ, providing new constraints on properties of cosmic neutrino sources.