Here is the contour plot of the self-distance surface. It shows the two pointy peaks of self-intersection. Any algorithm to find intersections will have to map the entire surface, which is intractible if this means evaluating each point on the surface. It does this by quickly examining the convex hulls of the curves over whole intervals of time, and discarding the intervals from further consideration when the convex hulls do not overlap. As the peaks get flatter, areas must be mapped instead of discarded.
Peaks become flat whenever curves approach each other in a grazing fashion, instead of incisively.
Instead of reporting such a peak once, bisection will report the intersection many times as it slowly maps the area of the swamp. Such multiple reports cannot be resolved by simply observing their closeness, because truly distinct peaks can be arbitrarily close together. Here is a final example that exhibits such a swamp. The two crossing points result in two peaks, which are at distinct but close positions near the middle of the crescent. Again, the resolution of the diagram is not sufficient to visibly separate the dual peaks.