In his answer to this MO question, Karl Schwede claimed that every non-normal variety can be obtained by an appropriate pushout diagram, as sketched in that answer. This would give substance to the heuristics according to which "a normal variety is a variety that has no undue gluing of subvarieties or tangent spaces" (again, see K.Schwede's answer and the examples therein).

Q. Has a proof of that fact been written down somewhere since then? If yes, where? And if not, could anybody who knows it sketch it here on MO?