You have

• A hidden minority (people don't know that the new anti-sodomy law is going to get Jimmy down the street killed)
• A very small minority (less power by sheer numerical disadvantage alone)
• A minority spread somewhat evenly throughout the population (similar to above, no localized power, no "strongholds", dependent and integrated with their majority peers)
• An easily demonized minority ("they'll turn your children/seduce your sons/daughters!")
• A minority you can subject to salacious moral outrage ("they lick each other's bums!", "sex is only for procreation!")

And with gays and lesbians you get all of that in one. What a deal! If you need a scapegoat for the newest punishment from God (read: plague, earthquake, whatever disaster happened last), you can point at them. Have a firebrand preacher yell to the masses that the toleration of this sinful behavior has brought the wrath of god onto the whole city. If you eradicate these sinners god will again smile on the good and righteous people of [insert whatever medieval hellhole you exist in].

Then you kill a bunch of them and go back to counting your tax bounty and donations from your god-fearing subjects. Keep up occasional persecutions, intensify as needed if you need to shore up your rule.

A playbook as old as time, it's just the details that vary. Currently there's a bit of a lull (in the west), thank god, but don't think religious fanatics of all sorts wouldn't go right back to it, if they could. And so the obsession simmers for future use.