If this unit has IE 8, they may be utilizing the "In Private Browsing" which does not save history.
The feature is not available when "Parental Control" is enabled.
IE (all flavors) must have 'Allow active content to run files in my computer' ; find in Tools> Int. options> Advanced
Firefox requires the Mozilla specific Flash enabled.
Flash will require 'Administrator' level account to run.
It may be a false positive, and it may be a valid block; a meta file exploit has lately reared it's ugly head and of course Windows may be vulnerable.
Make sure your Microsoft updates are current.
Let Norton expire; it's rubbish to begin with.
Try Avast (freeware; home version). Lightweight...