Downloaded apps don’t save to iPhone home screen

You mean the App Library I assume?

Settings > Home Screen, check Add to Home Screen

Also, you can move them out from the App Library onto the Home Screen:

From your Home Screen, swipe left until you get to the App Library.

Swipe downward and you'll get an alphabetical list of your apps.

Locate the app in the list or type the app name into the search field

Once you've located the app, tap/hold then drag it and it'll pop out to your Home Screen

Organize the Home Screen and App Library on your iPhone - Apple Support

It is definitely on a page of your Home screen if you downloaded it. Newly downloaded apps will be placed on the first available space on the first page of the Home screen that has an open space. They are never put into folders. If all of the pages of your Home screen are full it will still be placed in the App Library, which is the last page of your Home screen, and you can find it on that page by tapping Search to show the alphabetical list of your apps.

Lawrence Finch wrote:

It is definitely on a page of your Home screen if you downloaded it. Newly downloaded apps will be placed on the first available space on the first page of the Home screen that has an open space.

Hi Lawrence!

FYI, if Settings > Home Screen > Newly Downloaded Apps has App Library Only checked.. downloaded apps will not appear on your Home Screen. I believe Setting came with iOS 14 & the App Library

Good point. But I believe that it defaults to show on Home screen, so unless the OP changed the default setting it should be on the Home screen. But that’s clearly something to check.