History of downloaded apps

Read this support article for directions about how to see your purchase history in the App Store or iTunes Store [mobile device, computer, or web] --> View your purchase history for the App Store and other Apple media services - Apple Support

but also read the article about hiding and unhiding apps.

Incidentally:

Make sure that each family member has a unique Apple ID as suggested in this support article --> Make sure that each family member has a unique Apple Account - Apple Support

For a child (age varies according to country), read this support article about how to create an Apple ID for your child --> Create an Apple Account for your child - Apple Support

I don't know if you would have to do this from their account.

Hide purchases from the App Store - Apple Support

Free apps still count as purchased, they just don't charge anything.

I have never looked at my purchase history. I can't see any reason why they would show re-downloads.