Recent experiments on Majorana fermions in semiconductor nanowires [S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth, T. S. Jespersen, J. Nygård, P. Krogstrup, and C. M. Marcus, Nature (London) 531, 206 (2016)NATUAS0028-083610.1038/nature17162] revealed a surprisingly large electronic Landé g factor, several times larger than the bulk value - contrary to the expectation that confinement reduces the g factor. Here we assess the role of orbital contributions to the electron g factor in nanowires and quantum dots. We show that an L·S coupling in higher subbands leads to an enhancement of the g factor of an order of magnitude or more for small effective mass semiconductors. We validate our theoretical finding with simulations of InAs and InSb, showing that the effect persists even if cylindrical symmetry is broken. A huge anisotropy of the enhanced g factors under magnetic field rotation allows for a straightforward experimental test of this theory.