A quick check with Mathematica seems to suggest that a (smallest) counterexample is given by a graph with 6 vertices and 13 edges:
a sequence of edge removals http://www.nbi.dk/%7Ebudd/images/graphs2.png
Here the red edges are contained in the most odd cycles (16, 12, 6, 5, 2, 2, 1, 0 respectively) and one possible sequence of edge removals is shown. However, choosing different edges one can end up with a bipartite graph with 7 edges.
Edit: Any sequence of edge removals of the following graph with 8 vertices and 18 edges seems to result in a bipartite graph with at most 8 edges: a sequence of edge removals http://www.nbi.dk/%7Ebudd/images/graphs3.png
