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We study the carrier-density-dependent recombination dynamics in m-plane InGaN/GaN multiple quantum wells in the presence of n-type background doping by time-resolved photoluminescence. Based on Fermi's golden rule and Saha's equation, we decompose the radiative recombination channel into an excitonic and an electron-hole pair contribution, and extract the injected carrier-density-dependent bimolecular recombination coefficients. Contrary to the standard electron-hole picture, our results confirm the strong influence of excitons even at room temperature. Indeed, at 300 K, excitons represent up to 63 +/- 6% of the photoexcited carriers. In addition, following the Shockley-Read-Hall model, we extract the electron and hole capture rates by deep levels and demonstrate that the increase in the effective lifetime with injected carrier density is due to asymmetric capture rates in presence of an n-type background doping. Thanks to the proper determination of the density-dependent recombination coefficients up to high injection densities, our method provides a way to evaluate the importance of Auger recombination.