Such a reputation is probably mainly because the most anti-LGBT ones are the noisiest ones in news and politics relating to that issue.
http://www.pewforum.org/2016/05/12/changing-attitudes-on-gay-marriage/ indicates that majorities of Catholics and white mainline Protestants now favor same-sex marriage (and have done so since 2011). But it remains very unpopular with white evangelical Protestants, who make much of the noise on the issue. It is unpopular with black Protestants as well, though they make much less noise on the issue.