Actually, traditionally it wasn't a big deal for Jews to eat non-Jewish, non-supervised bread. This was because it was so simple, and very reasonable to assume what was in a bakery. But, with the advent of mass production in bakeries, and popularity of making various types of breads with fillings, we can't make that assumption anymore. The problem with the ingredients now is, largely, that milk products are generally being used in bread products. And, dairy is something that is more strict (chalav yisrael).
So, we used to be able to assume that non-Jewish bread was kosher. Not true, anymore.