We study the critical behaviour of the three-dimensional U(1) gauge+Higgs theory (Ginzburg-Landau model) at large scalar self-coupling lambda (``type II region) by measuring various correlation lengths as well as the Abrikosov-Nielsen-Olesen vortex tension. We identify different scaling regions as the transition is approached from below, and carry out detailed comparisons with the criticality of the 3d O(2) symmetric scalar theory.