First - not everyone does therapy on a weekly schedule. If you can't afford $60 a week, can you afford $60 every two weeks? $60 a month? I know very few people who go weekly.
Second - I did therapy online for the purpose of referrals, but my therapist was really easy to talk to. It was via text chat, so I can understand if that feels too impersonal but it was also $40 per session and he's very experienced with trans patients. His name is Graham Peveller if that sounds like it could work.
Last - depending on where you live, if there are universities or colleges nearby, you may be able to get free or low cost therapy if you're willing to work with a student. They wouldn't be experienced, but you'd essentially be helping each other.
Ultimately, copay would be the same no matter who you go to, unless you go the out of pocket route. If there are therapists you like, ask them what their out of pocket payment rate is. Your copay is typically going to be lower but you never know. You may be able to work something out if you tell them your copay is not something you can afford.
Not wanting to make you feel bad, but you have to understand what you're asking for. An experienced therapist has had at least 6 years of higher education that they paid for, professional licenses that need to be renewed, and they likely have to pay rent on their office space. Taking patients for $20/hour would be charity.