We report on the changes in mean square charge radii of unstable neon nuclei relative to the stable Ne-20, based on the measurement of optical isotope shifts. The studies were carried out using collinear laser spectroscopy on a fast beam of neutral neon atoms. High sensitivity on short-lived isotopes was achieved thanks to nonoptical detection based on optical pumping and state-selective collisional ionization, which was complemented by an accurate determination of the beam kinetic energy. The new results provide information on the structural changes in the sequence of neon isotopes all across the neutron sd shell, ranging from the proton drip line nucleus and halo candidate Ne-17 up to the neutron-rich Ne-28 in the vicinity of the "island of inversion." Within this range the charge radius is smallest for Ne-24 with N = 14 corresponding to the closure of the neutron d(5/2) shell, while it increases toward both neutron shell closures, N = 8 and N = 20. The general trend of the charge radii correlates well with the deformation effects which are known to be large for several neon isotopes. In the neutron-deficient isotopes, structural changes arise from the onset of proton-halo formation for Ne-17, shell closure in Ne-18, and clustering effects in Ne-20,Ne-21. On the neutron-rich side the transition to the island of inversion plays an important role, with the radii in the upper part of the sd shell confirming the weakening of the N = 20 magic number. The results add new information to the radii systematics of light nuclei where data are scarce because of the small contribution of nuclear-size effects to the isotope shifts which are dominated by the finite-mass effect.