We present a methodical study of grazing-incidence small-angle x-ray scattering performed in situ during pulsed-laser deposition of Pt on sapphire substrates. From measured two-dimensional intensity distributions in reciprocal space we calculated horizontal and vertical intensity projections and compare them to numerical simulations. The structure of the Pt layers was described using a simple Monte-Carlo model and the Ornstein-Zernicke theory with the Percus-Yevick approximation, and the scattering process was simulated using distorted-wave Born approximation. The validity of the structure models as well as the effect of the indirect scattering processes are discussed. From the comparison of the measured and simulated data we determined the lateral and vertical sizes of surface islands, the surface coverage, the island coalescence, as well as the thickness of the wetting layer at the substrate surface. We studied the time evolution of these parameters and their dependence on the substrate temperature.