In-beam gamma-ray spectroscopy using fragmentation reactions of both stable and radioactive beams has been performed in order to study the structure of excited states in neutron-rich oxygen isotopes with masses ranging from A=20 to 24. For the produced fragments, gamma-ray energies, intensities, and gamma-gamma coincidences have been measured. Based on this information new level schemes are proposed for O-21,O-22 up to the neutron separation energy. The nonobservation of any gamma-decay branch from O-23 and O-24 suggests that their excited states lie above the neutron decay thresholds. From this, as well as from the level schemes proposed for O-21 and O-22, the size of the N=14 and 16 shell gaps in oxygen isotopes is discussed in the light of shell-model calculations.