It’s no surprise that crit sessions, or critiques, are stressful. Everything about them, from the name on down, is designed to expose all of the things the artist/designer is doing wrong or what could be done better. An artist spends a day, a week, a month or longer developing an idea and it culminates in a session devoted to exposing their weaknesses. That sucks. Here are a couple things that will mitigate the stress.
As the artist, collect feedback consistently while you’re working so you aren’t thrown any curveballs on the day of the critique. Do self-critique, use your friends and colleagues for gut-checks while you're working and strive to anticipate feedback.
As a critic, tell the artist what could be made better but also tell them everything they did well. Be very clear about this. The best outcome of a critique is an artist who can identify weaknesses but can also replicate success.