Like.... when does the left not win the culture war? Progressives fought to end slavery, women's right to vote, etc. while conservatives fought against all of those things because they're inherently opposed to change.
I would say that the "born-again Christian" movements in the 80s and 90s showed the GOP that they could have an impact on culture. Nancy Reagan's "Just say no" campaign, as well as "The war on drugs". All these things that were championed by the GOP, were accepted by the main stream, served as a pretext for the GOP to continue to fight to influence American culture.
Of course, you're right, but I mean long term. As we've seen in this past decade gay rights have become the norm when in 2008, 2004, and 2012 ballot measures passed in several states banning gay marriage. The acceptance of cannabis use throughout society, so you're right as in they were once accepted as main stream, but slowly the left is winning another issue in the culture war.