Apps I thought were free are showing as purchased on iPhone

As far as the App Store is concerned, 'Purchased' is more analogous to 'downloaded'. The distinction is based on the fact that you have a license to install and use the app. The cost of the app (whether free or $1m) is irrelevant in this context.

The only question is whether you have been charged anything for installing the app. For that you can check https://apple.com/bill which will list all charges on your Apple Account.

Note that many apps are free to install, but offer in-app purchases after the fact. These, of course, would be charged (if you incurred them), despite the app initially being free. The App Store indicates if any given app supports in-app purchases.

You appear to have downloaded apps that offered a free trial. At the end of the trial period, if you have not properly canceled the subscription, you will start being charged for the subscription.

If you want to cancel a subscription from Apple - Apple Support

You might need to scroll down to find the Cancel Subscription button.

Scroll further down the link for instructions for non-Apple devices.

Things about subscriptions;

⁃ Deleting an app does not cancel the subscription.

⁃ A trial subscription must be canceled more than 24 hours before the renewal date or it may renew.

⁃ Many subscriptions are set to renew automatically until you cancel them.

⁃ If you see multiple subscription offers, as long as you don’t select one, the subscription shouldn’t renew.

⁃ If you don’t see Cancel Subscription, the subscription is canceled and won't renew.

⁃ An expiration date means the current subscription has been canceled but is available to use until that date.

⁃ Use Resubscribe/Renew if you change your mind about canceling the subscription.

If all else fails, check out Apple Subscription Support for articles and contact options;

https://getsupport.apple.com/?PGF=PGF63006&category_id=SC0060&symptom_id=23208

M1 Mac mini, macOS 15.2;  Watch, series 10 GPS-only, watchOS 11.2; iPhone 16, iOS 18.2; TV 4th Gen, tvOS 18.2; HomePods, audioOS 18.2 and Airport Express w/ Airplay 2

where are you seeing this? What were you charged? Free items appear on your previous purchases list.