Yes it's normal for a 6 year old to cry a lot and even in public some 6 year old children will cry a lot and throw tantrums especially in a store when they cannot get what they want.
I've seen 6 year old kids in stores that wanted things and the parents could not or would not buy what they wanted so they would cry and throw a tantrum in the store.
I saw one mother spank her 6 year old boy in the store over a tantrum and it stopped the tantrum right then and there although he still had tears in his eyes he stopped throwing the tantrum and was trying to stop crying.
All children are different and some 6 year old children may cry more than others and even I cried a lot when I was 6 years old.
But when a child gets above 6 years old they tend to cry less and by the time they get to 8 or 9 years old they will rarely if ever have a tantrum and cry in public in the store as they learn to control their emotions and learn that crying won't get them what they want.