What should I do if my preschooler wouldn't stop screaming because I didn't want to buy him a toy at the mall?

It is a matter of setting boundaries with him. You can talk to him before going for shopping today we are not buying any toy and we can buy only when it's your birthday. If he still throw tantrum at the mall, you should never give in. Let him throw tantrums in safe area until he settle down and explain to him the reason you are not buying.