Absolute photoionization cross sections for the ions N2+, N3+, O3+, O4+, F3+, F4+ and Ne4+ are measured using the merged-beam technique, combining the synchrotron radiation from an undulator at the storage ring ASTRID with ions produced by an ECR ion source. The spectral structures and the development of the cross sections along iso-electronic sequences are discussed. The experimental data are compared with R-matrix calculations from the Opacity Project and other sources, MCDF calculations, as well as the model functions provided by verner et al. (1996). In general, good agreement between experiment and calculations is observed for the magnitude of the continuum cross sections whereas very significant discrepancies are present for the resonance structures. The experimental data are also available in electronic form at the CDS.