Epitaxial CoFe2O4(CFO)/CoO bilayers were fabricated by pulsed laser deposition on flexible muscovite mica substrate. Samples with different CFO thicknesses were employed to study the phenomenon of exchange bias involving strongly anisotropic ferromagnet. Magnetic measurements exhibited great enhancement in the features of exchange bias. Raman and X-ray absorption spectroscopies indicated that a new phase emerged within the CFO layer because of the cation charge redistribution in CFO layer under bending, which in turn gave rise to anomalous hysteresis loops exhibited in the bent bilayers. These results provide a fundamental understanding about the mechanisms of exchange bias prevailing in these bilayers and call attention to the implementation of spintronic devices using flexible heterostructures such as the present CFO/CoO bilayers.