We report that the quasi-two-dimensional Fermi-liquid behavior of the spin-triplet superconductor Sr2RuO4 breaks down in the vicinity of the critical impurity concentration for the onset of magnetic order induced by nonmagnetic Ti4+ impurities. The non-Fermi-liquid behavior is interpreted in terms of the two-dimensional antiferromagnetic fluctuations, which arise mainly from the nesting within one of the Fermi-surface sheets. We argue against the main role of such magnetic fluctuations in the pairing mechanism.