Americans are evenly split over whether transgender people should use the bathrooms of their birth gender or of their current gender
In March 2016 North Carolina passed a bill which prohibits transgender people from using the bathrooms used by their current gender and instead required them to use the bathrooms of the gender they were born as, something at least seven other states are considering. The law is proving to be highly controversial as the state's Attorney General says that he will not defend the law, and large companies including PayPal have already canceled their plans to expand their presence in North Carolina as a result of the law.
Research from YouGov shows that Americans narrowly tend to favor (39%) rather than oppose (34%) laws which require transgender people to use the bathroom used by people of their birth gender, not their current gender, though over a quarter don't take a stance either way. Most Republicans (55%) and many independents (40%) support prohibiting transgender people from using their current gender's bathroom, but even 27% of Democrats agree. 46% of Democrats do think, however, that transgender people should be able to use the bathrooms of their current gender.
The split is exactly even when Americans are asked which bathrooms transgender people should use, with 37% saying they should use their birth gender and 37% saying their current gender. This split is due largely to people in the South, however. The South is the only region where people are more likely to say that transgender people should use their birth gender's bathrooms (41%) rather than their current gender's (30%). In every other region people tend to say that they should use their current gender's bathrooms.