The exchange bias of antiferromagnetic-ferromagnetic (AFM-FM) bilayers is found to be strongly dependent on the ferromagnetic spin configuration. The widely accepted inverse proportionality of the exchange bias field with the ferromagnetic thickness is broken in FM layers thinner than the FM correlation length. Moreover, an anomalous thermal dependence of both exchange bias field and coercivity is also found. A model based on springlike domain walls parallel to the AFM-FM interface quantitatively accounts for the experimental results and, in particular, for the deviation from the inverse proportionality law. These results reveal the active role the ferromagnetic spin structure plays in AFM-FM hybrids which leads to a new paradigm of the exchange bias phenomenon.