[238.5.2] Breakthrough Advance: .... A Breakthrough Advance is not Movement, per se; Advancing units spend no Movement Points, pay no terrain costs, and can move directly from one EZOC to another across Major River and Strait hexsides (i.e. it can grab a bridgehead; see 211.1).
I have a solidly manned major river line, but happen to have a few weak stragglers on the wrong side of the river. Why does a breakthrough give the enemy a free river crossing (bridgehead)? It doesn't make logic sense to me.