Hi R.,
I'm gonna pretty much agree with everyone else and tell you that the fits are absolutely normal and appropriate for age. They have the same feelings as adults - anger, fear, frustration, sorrow - but without the ability to control or express them like we do. Plus, they are beginning to exert their independence and "flex their muscles." This leads to lots of blow ups and tantrums and, yes, it can be frustrating and ,when in front of others, embarassing. The good news is that any parent worth their salt will know this is perfectly normal. For my part, when I am in public and hear a kid lose his sweet little mind, I usually say, "Oh, someone's not happy," to myself and move on.
You've gotten a lot of good advice here. I'll probably repeat some, but I thought I would share my "order of operations" that I use in order to avoid (and if that doesn't work) deal with conflict with my toddlers.
Try to state instructions of admonishments in a positive way like, "Please walk", "Use your quiet voice". Developmentally your child understands them more easily and can follow them better.
Give choices any time you can. Allow him to choose between two things as often as possible. It helps with the flexing muscles part because you are giving him appropriate ways to do it.
Redirection is awesome in avoiding conflict. Instead of laying down the law, so to speak, offer another alternative to the behavior you don't want. When he is running through the house determined to hurt himself, ask him to come help you fold washrags, or see if he wants to play beanbag toss with knotted up socks thrown into the laundry basket.
When my two young ones throw a fit, instead of starting with, "Stop" , I say, "I can't understand you when you are screaming or crying. Calm down so you can tell me what's wrong." This has worked really well. I guess just the validation that they will be heard helps.
If all else fails, I set up a spot that is just for fits. Not a time out really, but a place to cool off with some books, a blanket or lovey and such. I will say, "Do you need to go to your quiet place for a minute?" With sympathy, not censure.
Provide them with the words that they are having difficulty expressing. "Are you angry, tired, frustrated...." etc. And when you are having a play time act it out with him giving him lots of opportunities to practice appropriate responses.
Hope this helps, and try not to worry because they generally grow out of it.
L.