The new (Esri?) ground shape seems sufficiently detailed that it may be possible to nicely render bridge ways by drawing the way on an added planar surface grounded at the bridge ends. Seem to be 2 cases, with or without a man_made=bridge area:
if no man_made=bridge, treat each way (road, rail, path or whatever) independently, get the ground height at the bridge way ends (just where bridge ceases, not any intermediate divisions eg road name change points mid-bridge) and create a planar shape (grounded at the bridge ends, won't necessarily be exactly horizontal) wide enough to paint the way onto, and add the way texture on top of that rather than the ground.
with a man_made=bridge; create a planar shape using the man_made=bridge outline, work out where the 'ends' are (won't nec. be longer than it is wide, maybe the places where ways with matching layer value end near the perimeter) and get the ground heights for these 2 'ends', then paint all the way textures matching the man_made=bridge layer value onto this object.
(And if ground elevation is turned off, just skip all this)
Inevitably, it will get messy with special cases eg bridges with 3 'ends' etc etc but might be worth a go as I think it would often be impressive.
The new (Esri?) ground shape seems sufficiently detailed that it may be possible to nicely render bridge ways by drawing the way on an added planar surface grounded at the bridge ends. Seem to be 2 cases, with or without a man_made=bridge area: