We present TRANSITFIT1, a package designed to fit exoplanetary transit light curves. TRANSITFIT offers multi-epoch, multi-wavelength fitting of multi-telescope transit data. TRANSITFIT allows per-telescope detrending to be performed simultaneously with transit parameter fitting, including custom detrending. Host limb darkening can be fitted using prior conditioning from stellar atmosphere models. We demonstrate TRANSITFIT in a number of contexts. We model multi-telescope broad-band optical data from the ground-based SPEARNET survey of the low-density hot-Neptune WASP-127b and compare results to a previously published higher spectral resolution GTC/OSIRIS transmission spectrum. Using TRANSITFIT, we fit 26 transit epochs by TESS to recover improved ephemeris of the hot-Jupiter WASP-91b and a transit depth determined to a precision of 111 ppm. We use TRANSITFIT to conduct an investigation into the contested presence of TTV signatures in WASP-126b using 180 transits observed by TESS, concluding that there is no statistically significant evidence for such signatures from observations spanning 27 TESS sectors. We fit HST observations of WASP-43 b, demonstrating how TRANSITFIT can use custom detrending algorithms to remove complex baseline systematics. Lastly, we present a transmission spectrum of the atmosphere of WASP-96b constructed from simultaneous fitting of JWST NIRISS Early Release Observations and archive HST WFC3 transit data. The transmission spectrum shows generally good correspondence between spectral features present in both data sets, despite very different detrending requirements.