We report measurements of energy-dependent attosecond photoionization delays between the two outer-most valence shells of N$_2$O and H$_2$O. The combination of single-shot signal referencing with the use of different metal foils to filter the attosecond pulse train enables us to extract delays from congested spectra. Remarkably large delays up to 160 as are observed in N$_2$O, whereas the delays in H$_2$O are all smaller than 50 as in the photon-energy range of 20-40 eV. These results are interpreted by developing a theory of molecular photoionization delays. The long delays measured in N$_2$O are shown to reflect the population of molecular shape resonances that trap the photoelectron for a duration of up to $sim$110 as. The unstructured continua of H$_2$O result in much smaller delays at the same photon energies. Our experimental and theoretical methods make the study of molecular attosecond photoionization dynamics accessible.
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