Transition metal dichalcogenides (TMDs) are emerging as promising two-dimensional (2D) semiconductors for optoelectronic and flexible devices. However, a microscopic explanation of their photophysics, of pivotal importance for the understanding and optimization of device operation, is still lacking. Here, we use femtosecond transient absorption spectroscopy, with pump pulse tunability and broadband probing, to monitor the relaxation dynamics of single-layer MoS2 over the entire visible range, upon photoexcitation of different excitonic transitions. We find that, irrespective of excitation photon energy, the transient absorption spectrum shows the simultaneous bleaching of all excitonic transitions and corresponding red-shifted photoinduced absorption bands. First-principle modeling of the ultrafast optical response reveals that a transient bandgap renormalization, caused by the presence of photoexcited carriers, is primarily responsible for the observed features. Our results demonstrate the strong impact of many-body effects in the transient optical response of TMDs even in the low-excitation-density regime.