I can only answer from the liberal denomination point of view, this answer will not be accepted by all Christians.
The Old Testament has a number of restrictions that many of the arguments against homosexuality are drawn from. However, if a reader is honest with themselves when reading the Old Testament, then it’s impossible to come out of it without finding a number of verses that condemn the reader himself.
By the time of the New Testament there was a sect of scholars called the Pharisees who took a very legalistic view of the Old Testament law. Without getting too far into it, Jesus took exception with them because they understood the written law without understanding the reason for it. So even though they followed it letter for letter they completely missed the point.
And that’s the crux of the argument. Some churches think the Bible is the literal word of God, that every word is sacred and unquestionable. Other churches think the Bible is inspired - God’s word is in it, but the writers also pepper it with their own opinion and customs. Denominations that consider the Bible as inspired don’t give the “anti gay” verses as much thought. They consider them to be local custom, the writers opinion, or just another in a long list of faults that man has to deal with.