Statistical and multifractal properties of Barkhausen jumps in exchange-coupled ferromagnet/antiferromagnet bilayers are studied on Co/IrMn and NiFe/NiO heterostructures using a magneto-optical indicator film technique. The statistical analysis proves a nonstochastic character of magnetization jumps. In particular, power-law behavior is observed for Co/IrMn samples. Furthermore, the statistics depends on the ferromagnet layer thickness and antiferromagnet layer material structure. The average jump size displays an asymmetry between the forward and backward branches of the hysteresis loop, particularly pronounced for the structure with a thin Co layer. In spite of the observation of such an asymmetry in the activity of the domain nucleation and pinning centers, the statistical distributions of jumps do not show any significant differences for two branches of the hysteresis loop. The conclusion on a nonrandom character of the magnetization process is supported by the multifractal analysis which reveals the presence of correlations in the time arrangement of the Barkhausen jumps.