We present TransitFit, an open-source Python~3 package designed to fit exoplanetary transit light-curves for transmission spectroscopy studies (Available at https://github.com/joshjchayes/TransitFit and https://github.com/spearnet/TransitFit, with documentation at https://transitfit.readthedocs.io/). TransitFit employs nested sampling to offer efficient and robust multi-epoch, multi-wavelength fitting of transit data obtained from one or more telescopes. TransitFit allows per-telescope detrending to be performed simultaneously with parameter fitting, including the use of user-supplied detrending alogorithms. Host limb darkening can be fitted either independently (uncoupled) for each filter or combined (coupled) using prior conditioning from the PHOENIX stellar atmosphere models. For this TransitFit uses the Limb Darkening Toolkit (LDTk) together with filter profiles, including user-supplied filter profiles. We demonstrate the application of TransitFit in three different contexts. First, we model SPEARNET broadband optical data of the low-density hot-Neptune WASP-127~b. The data were obtained from a globally-distributed network of 0.5m--2.4m telescopes. We find clear improvement in our broadband results using the coupled mode over uncoupled mode, when compared against the higher spectral resolution GTC/OSIRIS transmission spectrum obtained by Chen et al. (2018). Using TransitFit, we fit 26 transit observations by TESS to recover improved ephemerides of the hot-Jupiter WASP-91~b and a transit depth determined to a precision of 170~ppm. Finally, we use TransitFit to conduct an investigation into the contested presence of TTV signatures in WASP-126~b using 126 transits observed by TESS, concluding that there is no statistically significant evidence for such signatures from observations spanning 31 TESS sectors.
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