The origin of the TeV–PeV astrophysical neutrinos seen by the IceCube telescope is unknown. If they are made in proton-photon interactions in astrophysical sources, their spectrum may show bump-like features. We search for such features in the 7.5-years High-Energy Starting Events (HESE), and forecast the power of such searches using larger data samples expected from upcoming telescopes. Present-day data reveals no evidence of bump-like features, which allows us to constrain candidate populations of photohadronic neutrino sources. Near-future forecasts show promising potential for stringent constraints or decisive discovery of bump-like features. Our results provide new insight into the origins of high-energy astrophysical neutrinos, complementing those from point-source searches.