The electron Land\'e g factor ($g^{*}$) is investigated both experimentally and theoretically in a series of GaBi$_{x}$As$_{1-x}$/GaAs strained epitaxial layers, for bismuth compositions up to $x = 3.8$%. We measure $g^{*}$ via time-resolved photoluminescence spectroscopy, which we use to analyze the spin quantum beats in the polarization of the photoluminescence in the presence of an externally applied magnetic field. The experimental measurements are compared directly to atomistic tight-binding calculations on large supercells, which allows us to explicitly account for alloy disorder effects. We demonstrate that the magnitude of $g^{*}$ increases strongly with increasing Bi composition $x$ and, based on the agreement between the theoretical calculations and experimental measurements, elucidate the underlying causes of the observed variation of $g^{*}$. By performing measurements in which the orientation of the applied magnetic field is changed, we further demonstrate that $g^{*}$ is strongly anisotropic. We quantify the observed variation of $g^{*}$ with $x$, and its anisotropy, in terms of a combination of epitaxial strain and Bi-induced hybridization of valence states due to alloy disorder, which strongly perturbs the electronic structure.