We experimentally investigate the single and double ionization of N_{2} and O_{2} molecules in bicircular two-color femtosecond laser pulses, and compare with their companion atoms of Ar and Xe with comparable ionization thresholds. Electron recollision assisted enhanced ionization is observed in N_{2} and Ar by controlling the helicity and field ratio between the two colors, whereas the enhanced ionization via the recollision is almost absent in O_{2} and Xe. Our S-matrix simulations clearly reveal the crucial role of the detailed electronic structures of N_{2} and O_{2} on the two-dimensional recollision of the electrons driven by the bicircular two-color laser fields. As compared to Ar, the resonant multiphoton excitation dominates the double ionization of Xe.