we went thru this too.

the first few times are embarrassing, but if you have a practise/ work out what you're going to say - then stick to it - it does get easier.

toddlers are working out boundaries
"i wonder what happens if i throw a tantrum - will i get my own way"?
so i have a pat line for when this happens
"i am not going to reward naughty behaviour"
i keep repeating it
after a while, the behaviour starts to weaken, disappear, because your child comes to learn
"oh bum, i throw the tantrum, but don't get what i want, so no pay off, not worth it".

you will go thru some embarrassing moments in the shops
but many people who look, will be giving you sympathetic "yep, been there myself, you poor thing" type looks - cos i think EVERY parent goes thru it.

if you have the option (e.g a babysitter), NOT taking your child shopping is another thing to try e.g you will go with me again, when your behaviour improves. (if your child understands that).

i didn't use the naughty chair either. First i heard my local Child Youth HEalth recommended that method for children that are at least four, then by the time my child turned four, i read articles that made me think twice about the Naughty Step, actually being of benefit.

i avoid conflict, and when you have a child having a tantrum, becuase it's noisy and embarrassing, it's SO tempting, to just give in, to STOP the immediate embarrassment.
but it comes down to - can you handle the embarrassment a few times OR do you want to go broke, being blackmailed like this EVERY time you go shopping for the next 15 years?
Part of socialisation is teaching a child to accept the word "no". It is part of life. and you don't have to say it in a nasty way or anything. I look at what my child is wanting, i achnowledge what she is admiring (even if i can't stand it myself), and say something sympathetic to go with my "no".

hth