Ito, Moynihan, and Angell [Nature 398, 492 (1999)] compared the kinetic and thermodynamic measures of a glassformer's "fragility." For the liquids they considered which include molecular, covalent, and ionic substances, thermodynamic fragility and kinetic fragility follow the same order, suggesting that the thermodynamic data alone may be sufficient to determine the kinetic fragility of a liquid. Here we restrict the comparison to a large number of molecular glass-forming liquids, and find breakdowns of the proposed correlation. The absence of an immediate connection between thermodynamic and kinetic fragilities at least in molecular liquids may be due to the effect of kinetic factors and cooperative many-body molecular dynamics. The development of a relation between them requires taking into consideration the cooperative many-body molecular dynamics. The latter governs, in conjunction with thermodynamics, the kinetics of glass-forming liquids, and thus is the missing link between the two.