How to go about obtaining data
You should try contacting cities about getting the sensor data from their Traffic Controller Systems. These usually have several detectors per lane at intersections (they also control the traffic lights so you get data on those as well). Some of these data are property of the companies that supplies these system, in which case you'll need to contact the company. These companies are generally not keen on sharing the data with commercial parties, but generally are interested in sharing it with educational parties as outreach.

I know one public dataset of this kind of data, the city of Utrecht in the Netherlands provides a sample of traffic density data. Unfortunately the metadata is mostly in Dutch.