We have investigated the carrier dynamics in In0.53Ga0.47As/InP quantum wells with various well widths as a function of temperature and carrier density using time-resolved photoluminescence spectroscopy. At low temperatures (T<50 K), we find pure excitonic recombination. At elevated temperatures, two radiative recombination paths are present, and the proportion of free-carrier recombination to excitonic recombination increases as the temperature is raised. We observe a nonlinear increase of the radiative lifetime caused by thermal ionization of excitons. Exciton binding energies in the range between 9 and 15 meV and majority carrier concentrations of about 1.2 x 10(10) cm-2 are deduced from the ratio between the high and low injection lifetimes. The exciton binding energy increases with decreasing well thicknesses, in good agreement with theoretical predictions.