Could I get all the linearly independent cycles of length 4 if I iteratively cut one of the edges in the 4-membered cycles detected?

Question

I would like to get all the linearly independent 4-membered cycle in a graph. Suppose that there are $n$ linearly independent 4-cycles in the graph. If I cut one of the edges in a 4-cycle found, is it true that the remaining graph contains $n-1$ linearly independent cycles of length 4? Does anybody know a proof/counterexample? How could I generate all the linearly independent cycles of length 4 in a bipartite graph?