It looks like almost all bridges, regardless of the OSM highway tag type are rendering as if the tags are highway=primary and bridge=yes. See attached screen capture. There is one bridge shown here that is rendered correctly, the difference between it and the others is that it has a layer tag while the others do not.
Interesting discovery about the layer attribute. One of the more frustrating tags in OSM; the Wiki clearly states:
“Bridges should have a layer=, for simple crossings almost always layer=1″*
Yet despite this wording, many choose to ignore it and most discussions on the topic are dominated by those who feel the tag is redundant and that by nature bridge should imply that it’s “over” some other object and that renderers should also assume this. Thus they map the bridge and leave the default at level=0.