We implement an extremely broad second-harmonic spectrum of about 90 nm in width based on a 7-fs mode-locked Ti:sapphire laser. This broadband second-harmonic signal is used as the probe in a non-degenerate pump-probe experiment to investigate the ultrafast carrier dynamics in wide band-gap semiconductors. To properly calibrate the pump-probe data, the time delays between the pump of a particular wavelength and the probes of various spectral portions are determined through the interferometry measurement and the dispersion calculation. To demonstrate the pump-probe experiment operation, we measure the carrier relaxation process from the excitation levels down to the free-carrier and the localized states in an InGaN thin-film sample, in which indium-rich nano-clusters exist to form the localized states. From the time-resolved differential transmission profiles at various spectral positions of an infinitesimal spectral width and the temporal evolution of probe spectrum, one can observe the following relaxation process: First, once carriers are excited, only a small portion of carriers relaxes into the free-carrier and localized states independently within 1 ps. Then, the major part of carriers starts to relax into the two groups of states not until several ps after excitation. Such a relaxation process does not seem to be cascading, i.e., relaxation into the localized states through the free-carrier states.