AHA!!! You've hit on something crucial, I think. In these people's minds, there is no conflict as they interpret Scripture themselves. The other day I was looking for an answer to the exact same question you have and I came across a site (didn't bookmark it though) which "explained" all the Scripture passages dealing with homosexuality in a completely different way than has been Traditional Judeo-Christian teachings for thousands of years. The ancient words were described as having different meanings and of course, Sodom and Gomorrah were destroyed for being "hostile to outsiders" and not for craven behaviour.
Makes me all the more appreciative of Scripture coupled with a 2,000 year old Tradition. Scripture coupled with personal interpretation is poison.
Now it makes sense. I was fed poison from all those personal priestly homilies growing up?
God's word is pure and simple and sharper than any 2-edged sword. It cut(s) my heart wide open (without any magisterium's assistance) to see what a lost, vile, religious sinner I was.