This paper proposes an iterative method based on the conjugate gradient method on the normal equations for finding the generalized bisymmetric solution X to the system of linear operator equations

where A₁, A₂, ..., A₁ are linear operators. By the iterative method, the solvability of this system over the generalized bisymmetric matrix X can be determined automatically. When the system of linear operator equations is consistent over the generalized bisymmetric matrix X, the iterative method with any generalized bisymmetric initial iterative matrix X(1) can compute the generalized bisymmetric solution within a finite number of iterations in the absence of roundoff errors. In addition, by the proposed iterative method, the least Frobenius norm generalized bisymmetric solution can be derived when a special initial generalized bisymmetric matrix is chosen. Finally, two numerical examples are presented to support the theoretical results of this paper.