It's highly unlikely that this data exists for all of the US, but certain cities or states may provide it.
Searching for "road intersections" on ArcGIS.com, I found that some state DOT's (Departments of Transportation) provide point data for intersections. Eg, Florida DOT provides a layer with very minimal attributes, and nothing about left turn signals.
West Virginia DOT provides a feature layer with "Large Arrows Two Directions, Turn Lane Signs, Advance Arrows," at least according to the layer description. The actual feature data looks like this:
So...maybe the left turn arrow information is in "MUDCDCODE" or "SIGN_SYS"...your guess is as good as mine. There's probably an explanation somewhere on the WVDOT website.
Once you've looked at all 50 states, you might want to try looking for data from the larger cities in the US. Don't bother with small towns, frankly they're doing well if they even have a website.
Some of the more affluent/densely populated counties also provide some geospatial data, although you'll probably catch this when you search city-level.