Accurate physical parameters of exoplanet systems are essential for further exploration of planetary internal structure, atmospheres, and formation history. We aim to use simultaneous multicolour transit photometry to improve the estimation of transit parameters, to search for transit timing variations (TTVs), and to establish which of our targets should be prioritized for follow-up transmission spectroscopy. We performed time series photometric observations of 12 transits for the hot Jupiters HAT-P-19b, HAT-P-51b, HAT-P-55b, and HAT-P-65b using the simultaneous four-colour camera MuSCAT2 on the Telescopio Carlos Sánchez. We collected 56 additional transit light curves from TESS photometry. To derive transit parameters, we modelled the MuSCAT2 light curves with Gaussian processes to account for correlated noise. To derive physical parameters, we performed EXOFASTv2 global fits to the available transit and radial velocity data sets, together with the Gaia DR3 parallax, isochrones, and spectral energy distributions. To assess the potential for atmospheric characterization, we compared the multicolour transit depths with a flat line and a clear atmosphere model. We consistently refined the transit and physical parameters. We improved the orbital period and ephemeris estimates, and found no evidence for TTVs or orbital decay. The MuSCAT2 broad-band transmission spectra of HAT-P-19b and HAT-P-65b are consistent with previously published low-resolution transmission spectra. We also found that, except for HAT-P-65b, the assumption of a planetary atmosphere can improve the fit to the MuSCAT2 data. In particular, we identified HAT-P-55b as a priority target among these four planets for further atmospheric studies using transmission spectroscopy.