Intense-laser ionization rates for rare gas atoms and diatomic molecules have been precisely compared by making simultaneous measurements of ionization yield vs laser intensity for mixed atomic and molecular targets. At a given laser intensity, the N (2) and F (2) ionization yields are slightly greater than that of Ar. Conversely, comparison of O (2) and S (2) with Xe indicates significant ionization suppression in these molecules. Recent molecular ionization models that successfully describe ionization suppression in O (2) and its absence in N (2) fail to explain our observations in F (2) and S (2).