The propagation of spin-wave beams in ferromagnetic stripes has been investigated experimentally by means of spatially resolved Brillouin light scattering spectroscopy. It is shown that in the case of low-amplitude spin waves the diffraction and the transverse confinement of spin waves lead to oscillations of the beam width. The authors found that with increasing spin-wave amplitude the interplay between the linear diffraction and the repulsive spin-wave nonlinearity results in a suppression of the oscillations of the beam width and in its transverse stabilization.