I worked as part-time assistant at college, and everything was going well at first. However, something went wrong. I screwed up a piece of work and now teacher is extremely upset at my bad performance. это звучит нормально?

Don't forget to add a/my before teacher, otherwise we don't know what teacher you're talking about.

"...and now my/a teacher is extremely..."

And "upset at my bad performance" is something everyone can understand, but since you were speaking informally before ("screwed up"- kudos to you for learning that slang) it would sound better if you said something like, "and now my teacher is very mad at me."

Or, since you already said "screwed off", you could say "and now my teacher is super pissed off at me." (More informal slang that is at a similar formality level of "screwed up".)