I've been in your situation at least a dozen times. It sucks. I developed chronic anxiety over those incidents. It combines public humiliation, disappointment of your peers, and sometimes disciplinary action all into one thing. If you have co-workers that are more backstabbers than supporters it's even worse.
The only thing I can recommend is when you make a mistake, own it 100% and do everything you can to make it right. In fact- OVER own it. Make it clear that this isn't the level of work you expect of yourself and you're going to do everything you can to rectify it. When you act like this, you'll find other people start coming to your defense and helping. This is opposed to giving excuses and blame like "I had too many things going on" or "Dave didn't do what I told him" or "Larry messed it up, not me."
Once you end up working to correct your mistake, you'll also find you never make it again, and inevitably learn some new skills and solutions along the way. A mistake is a learning experience. If you don't improve from it, it's a failure.
Also remember the perception of yourself in your head is always 100x more critical than the way others perceive you. It took some therapy and medication for me to realize that.