The width=* key is defined as specifying things like the width of a way. See http://wiki.openstreetmap.org/wiki/Key:width
However I haven't seen a map renderer that honors that key or the lanes=* key. And you are correct, at some zoom levels it can result in streets that overlapping other features.
I would suggest that you assure the tagging is there to properly specify the width so when the renderer(s) are updated to honor the tagging it will show correctly.
The style for the default map at http://www.openstreetmap.org/ is maintained at https://github.com/gravitystorm/openstreetmap-carto and I believe there is a page there for reporting and tracking issues like this.