We study the anisotropic dynamic formation of photo-induced waveguide in biased photorefractive media by means of a time resolved three-dimensional numerical model of the space-charge field formation. Specific features of the dynamic creation of 1D-confined planar and 2D-confined channel waveguides are elucidated and confirmed experimentally. Thanks to a transient space-charge field overshooting in both 1D and 2D geometries, light confinement in the first dimension occurs at a very early stage in the formation of the space-charge field. In the 2D channel type waveguides, the final shape of the waveguide index contrasts and modes depend not only on the illumination geometry and intensity distribution, but also on the anisotropy of the charge mobility along the two transverse directions.