Traffic could cause roads: greater traffic leads to greater expenditure on roads. Or roads could cause traffic: greater supply of roads leads to more driving and hence traffic.  Which way is it?

This is studied by Duranton and Turner in a forthcoming paper in AER.  They use an instrumental variables approach to identify causation.  A 1947 plan envisaged a highway network connecting existing population centers.  Importantly for this study, tha plan was based on existing population centers not forecast traffic demand.  Hence, the impact of the greater availability of roads on traffic can be studied (while controlling for factors such as population).  The authors find:

For interstate highways in metropolitan areas we find that VKT (vehicle kilometers traveled)
increases one for one with interstate highways, confirming the ‘fundamental law of highway congestion’

Provision of public transit also simply leads to the people taking public transport being replaced by drivers on the road.  Therefore:

These findings suggest that both road capacity expansions and extensions to public transit are not
appropriate policies with which to combat traffic congestion. This leaves congestion pricing as the main
candidate tool to curb traffic congestion.