Charge extraction in organic solar cells (OSCs) is commonly believed to be limited by bimolecular recombination of photogenerated charges. However, the fill factor of OSCs is usually almost entirely governed by recombination processes that scale with the first order of the light intensity. This linear loss was often interpreted to be a consequence of geminate or trap-assisted recombination. Numerical simulations show that this linear dependence is a direct consequence of the large amount of excess dark charge near the contact. The first-order losses increase with decreasing mobility of minority carriers, and we discuss the impact of several material and device parameters on this loss mechanism. This work highlights that OSCs are especially vulnerable to injected charges as a result of their poor charge transport properties. This implies that dark charges need to be better accounted for when interpreting electro-optical measurements and charge collection based on simple figures of merit.