K.S.
I remember those days. In fact, I remember when my then-two year old threw himself onto the ground in the middle of a parking lot while it was raining and I was carrying an armload of dry cleaning. He was shrieking like a banshee. All I did was pick him up and move him to the sidewalk. Then I set him down and stood there watching him. I could have taken him back to the car, but that would have rewarded his tantrum so I was bound and determined to take my husband's uniforms into the cleaner no matter how long it took.
I stood there for a good five minutes. (And it felt like twice as long.) I had people approach and ask me if they could "help." I covered my extreme embarassment by politely and calmly saying, "No thank you. We're fine." I didn't threaten to walk away from my son. I actually didn't say a word to my son. I just stood there patiently and waited for him to shriek himself out. When he did, I calmly asked if he was ready to take Daddy's clothes into the cleaner. He shrieked "NO!" at the top of his lungs and began yelling like a banshee again. I just stood there waiting patiently again amd wondered if anyone had called the police or child protective services. :) Fortunately, no one had and my son quickly figured out that Mama wasn't going to cave in. He stood up and we walked into the dry cleaners like nothing had happened. Darn kid!
With my little guy, I tried to schedule errands during times he was not over-tired or hungry. I also kept in mind the fact that he didn't have alot of stamina or tolerance for a long day of errands.
I'm a big believer in facing the embarassment head on. Sure there are going to be people (who have never had kids) who will look at you like you're a terrible mother, but almost everyone will be looking at you with sympathy while thinking "thank goodness my kids are past that stage!) As long as you are calm and are keeping your daughter from hurting herself, let Amelia have a fit or a meltdown. Don't cave in. Don't reward her bad behavior because if you cave in while you're in public she will quickly figure out how to push your buttons! But if you don't reward her for tantrums, she'll stop having them.
Just remember: You are not alone. Your daughter is not a brat. She's just TWO. You have no reason to be embarassed when she has a meltdown! (Unless she's still doing it when she's six!) And every other parent has been there at some point and they're staring at you because they're glad they don't have a two year old! (In another year, I will have TWO boys turning two. And I am NOT looking forward to those testing tantrums!)