Spin−orbit torque (SOT)-induced magnetization switching in ferrimagnetic materials is promising for application in a new generation of information storage devices. Here, we demonstrate SOT-induced field-free magnetization switching of the perpendicularly magnetized CoTb ferrimagnet layer in the IrMn/CoTb bilayer, in which the in-plane magnetic inversion symmetry is broken by a spontaneous in-plane exchange bias (IEB) established by isothermal crystallization of the IrMn layer. We obtain a significant SOT effective field acting on the CoTb layer and a large effective spin Hall angle in this system, derived by the second harmonic voltage measurement method. Moreover, the IrMn/CoTb bilayer demonstrates multistate magnetic switching behavior with different amplitudes of current pulses at zero field, which can mimic the synaptic weight updates in the neuromorphic network. These findings make the IrMn/CoTb bilayer with spontaneous IEB a promising candidate for potential applications in multilevel storage and neuromorphic computing.