There are more than 200 moons in our Solar System, but their relatively small radii make similarly sized extrasolar moons very hard to detect with current instruments. The best exomoon candidates so far are two nearly Neptune-sized bodies orbiting the Jupiter-sized transiting exoplanets Kepler-1625 b and Kepler-1708 b, but their existence has been contested. Here we reanalyse the Hubble and Kepler data used to identify the two exomoon candidates employing nested sampling and Bayesian inference techniques coupled with a fully automated photodynamical transit model. We find that the evidence for the Kepler-1625 b exomoon candidate comes almost entirely from the shallowness of one transit observed with Hubble. We interpret this as a fitting artefact in which a moon transit is used to compensate for the unconstrained stellar limb darkening. We also find much lower statistical evidence for the exomoon candidate around Kepler-1708 b than previously reported. We suggest that visual evidence of the claimed exomoon transits is corrupted by stellar activity in the Kepler light curve. Our injection-retrieval experiments of simulated transits in the original Kepler data reveal false positive rates of 10.9% and 1.6% for Kepler-1625 b and Kepler-1708 b, respectively. Moreover, genuine transit signals of large exomoons would tend to exhibit much higher Bayesian evidence than these two claims. We conclude that neither Kepler-1625 b nor Kepler-1708 b are likely to be orbited by a large exomoon.