We study experimentally the propagation of nanosecond spin-wave pulses in microscopic waveguides made of nanometer-thick yttrium iron garnet films. For these studies, we use micro-focus Brillouin light scattering spectroscopy, which provides the possibility to observe propagation of the pulses with high spatial and temporal resolution. We show that, for most spin-wave frequencies, dispersion leads to broadening of the pulse by several times at propagation distances of 10 micrometers. However, for certain frequency interval, the dispersion broadening is suppressed almost completely resulting in a dispersionless pulse propagation. We show that the formation of the dispersion-free region is caused by the competing effects of the dipolar and the exchange interaction, which can be controlled by the variation of the waveguide geometry. These conclusions are supported by micromagnetic simulations and analytical calculations. Our findings provide a simple solution for the implementation of high-speed magnonic systems that require undisturbed propagation of short information-carrying spin-wave pulses.