Bose-Einstein condensates of exciton-polaritons in inorganic semiconductor microcavities are known to possess strong interparticle interactions attributed to their excitonic component. The interactions play a crucial role in the nonlinear dynamics of such systems and can be witnessed as the energy blueshifts of polariton states. However, the localised nature of Frenkel excitons in strongly coupled organic microcavities precludes interparticle Coulomb exchange-interactions that change mechanisms of the nonlinearity and blueshifts accordingly. In this report, we unravel the origins of blueshifts in organic polariton condensates. We examine the possible contributions: intracavity optical Kerr-effect, gain-induced frequency-pulling, polariton interactions and effects related to saturation of optical transitions for weakly- and strongly-coupled molecules. We conclude that blueshifts in organic polariton condensates arise from the interplay of the saturation effects and intermolecular energy migration. Our model predicts the commonly observed step-like increase of both the emission energy and degree of linear polarization at the polariton condensation threshold.