The turrets don't really have items. They were visually given 'item' slots and given 'items', but those items are just descriptions of the passives the turrets normally has (for instance, damage ramps up, or turrets have more resistances when no enemy minions are nearby).
Actually buying items for turrets would likely warp the game in a pretty bad way. The problem is putting gold in a turret is a waste unless that upgrade is worth more then an actual item for your champion OR you are full build and have nothing else to do. The latter means the players most likely to be able to upgrade turrets are the ones winning and able to freely farm...which means controversially that the team that needs the least help defending will have the strongest turrets, while the team that most needs help defending will need to either gimp their champions or have weaker turrets. This runs pretty contrary to most of Riot's goals (where lategame every team is on a relatively even footing given enough time).