Icons do not appear on Home Screen after I have downloaded the apps

On what device is this happening? What is its version system software (OS)?

What apps did you try to get?

Downloaded apps will appear on the device on which you used to App Store app, not on other devices.

On iPhone, new icons may appear on secondary or later pages of the home screen, if you swipe sideways.

Try this: In the App Store app, go to the Today tab, and find your account icon circle in the top-right corner. Click that to see you account page for the App Store. There will be an option “Purchased”, click that. That will open a page with a list of apps that you have downloaded on any of your Apple devices (if also available for iPhone), in reverse order. Do you see your ‘missing’ recently downloaded apps there? Does it have an ‘Open’ button next to it? Then the app is somewhere on your device.

If you do not see the apps there, then they are not on your phone and never were. Possibly an interrupted download?

If you do see the apps in the list, but with a cloud icon next to it, the they were purchased apps, but not currently on that iPhone.