You're kids throw a fit in public. They scream in that old lady's ear. They bang into that other mom while running from you. They're just generally be bad and uncontrollable. You're embarrassed. Again.
Take them outside (obviously). Calm them down (obviously). Explain to them why certain behavior is not acceptable in certain situations and why other behavior is not acceptable ever (obviously).
Then take them back in, and don't pretend like nothing happened. That's not how real life worked, and at four (because my kids are four now), I expect them to start understanding that their actions have an effect on other people. I make mine apologize, if someone has suffered in some personal way due to their bad behavior. So the old lady and the other mom would get an 'I'm sorry,' from my girls. Does this work to stop the behavior in the future?
No. Not even close. But that's okay. It's not meant to shame them into being good, after all. It's meant to show them that you have to take responsibility for yourself, even when you've flown off the handle. That things you express in anger still count in the real world, even though you were angry at the time. It shows them that after the anger ebbs, the bruises remain.
Better still, usually the people being apologized to are quite gracious. They thank the girls for the apology and tell them they are nice girls, or what have you. This shows that understanding you were wrong and making amends doesn't change what happened but does make a difference to those involved.
So, it's not an immediate fix. It's a slow-working introduction to how to be a human being.