> How exactly [can LGBT+ people be hurt by conversion therapy]?
Great question. I'm going to assume this is being asked in good faith.
If someone feels attraction to same sex people, and they are told by important people in their lives that this is shameful, they can internalize that shame. This can lead to thoughts like, "I feel bad about who I am", "I am disgusted by who I am", "Why can't I escape these feelings about who I am?!", "No one will ever love me if I am like this.", "I have to stop these feelings or I'll be evicted from my community.", etc.
Constant internal monologues like this can create stress, anxiety, and depression. Adding a religious element can increase these feelings. If someone is spending a significant amount of time wondering, "Am I going to be eternally punished if I can't shake these feelings?" I can imagine that could cause serious, significant, and lasting anxiety.
Conversion therapy, in my understanding, reinforces these feelings. Telling someone, "yes, it is shameful to feel this way. Yes, you are at risk of eternal damnation if you act on these feelings." It's important to differentiate between Conversion Therapy and Therapy.
Shame, anxiety, depression, and stress like this lead to all sorts of negative health outcomes -- depressed immune systems, negative social outcomes, self-harm, suicide, etc.
It's important to understand that I'm asserting just that this can cause harm. Not that it will always cause harm. I'm specifically pushing back on the idea that if Conversion Therapy doesn't involve physical abuse, it therefore cannot cause harm.