Hi! I'm a member of the Church of Jesus Christ of Latter Day Saints, so hopefully I'll be able to shed some light on this for you.
Youth (usually on Tuesday) nights are get-togethers of the youth of the church, friends are invited and welcomed. Activities can be different every week, but tend to be fun with a purposeful activity (rather than just hanging around). For example, over the last few weeks the kids at my local church have made pizza, played mini-golf with holes they had designed themselves out of cardboard and pipes, and done yoga and quizzes.
Religiously speaking, they started and finished with a prayer, by which I mean everyone stood around while one person offered a 30 second prayer saying thanks for the activity and please help everyone get home safely (etc).
The activities will be different every week, so one night they might be playing hockey, but then the next week they might be working on a more spiritually-focussed goal or doing a service activity. When my kids want to invite friends, they usually pick and choose (so say "tonight it's football, can we invite Billy") etc. The aim is to help them have friendships and a nice time, but also to help their spirituality, so there will be a balance of activities.
I can understand how anyone who only knew the church by reputation would be hesitant, truly. The common assumptions are clear to see on this thread. I've found 'youth' to be really positive and constructive, they just have a good time, they're supervised, no-one drinks or smokes, the adults are clued-up on safeguarding and the activities are pretty wholesome and fun. They seem to get the good part of tween/teenage friendships without the pressure they're under at school, if that makes sense.
If you have any questions, please feel free to pm me :)