There's bad news and good news. The bad news is that you cannot wish tantrums away and, yes, your kids are probably going to embarrass you in public on any number of occasions. That's, unfortunately, part of parenting.
The good news is this: If you learn how to think about tantrums correctly and realistically, and if you learn how to handle these outbursts calmly and decisively, you can train most kids to outgrow this obnoxious phase of their existence. Some children may take a few days, some a few months, and others may still blow up from time to time, but at a rate that is way less than the tantrum frequency they started at.