There is a short answer, and a longer answer. 

The short answer is that Christ calls us to love our neighbor. Jesus makes no claims that this doesn't apply to LGBTQ+ folks, so neither do we.

The long answer is that scripture is always being interpreted and reinterpreted. Over two hundred years ago, many Christians were certain that slavery was morally appropriate, and used the bible to justify their greed and bigotry. The same style of argument is used today to deny rights and privileges to LBTGQ+ folks. We reject the use of the bible as a means of oppression regardless of the timeframe. Jesus never calls us to oppress others, only to liberate. 

That being said, it's actually pretty easy to refute the passages often used to clobber folks who are LGBTQ+; we ignore the teachings of Leviticus wholesale (ex. prohibition on eating pork, requirements of fences on roofs, teachings on slavery, punishment, and debt forgiveness) so it's hypocritical to point to two passages (18:22 & 20:13) as authoritative. Romans 1:26-27 ends with a claim that judging others is equally if not more sinful than anything listed (see Romans 2:1). 1 Corinthians 6:9 lists people who is use violence against others, when read in a sexual sense it becomes clear this has nothing to do with loving relationships. Moreover, those who argue for the oppression of LGBTQ+ people totally ignore St. Paul's teaching that we are called to celibacy first, and that marriage should be reserved for those who are already married. 

We are not interested in using the bible as a cudgel to maim our enemies, we are interested in using the bible as a tool to open our hearts so that we might love our enemies (Matthew 5:44). That is the teaching of Christ, not a gospel of oppression.