Updating iphone 5S 10.3.4. and can't download apps

You need a newer (used, refurb) or new iPhone, if apps you want or need require need iOS 13, or iOS 14, or the upcoming iOS 15.

Your iPhone 5 cannot be upgraded to iOS 11 or later.

The iPhone 5 lacks the necessary hardware.

There are no hardware upgrades.

Oldest iPhone that can run iOS 14.7.1 (current) is iPhone 6s.

Here is what Apple,suggests ➡️ If you can’t download or update apps on your iPhone or iPad - Apple Support

An unit running 10.3.4 isn't capable of 11 or above. If you encounter a specific issue downloading applications compatible with it, describe it in this thread.

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teri555 wrote:

I tried everything in the Apple article and restarted yet again, but no change.

Might have missed step 6 from the article?

6: Contact Apple

If you still can't download or update apps on your iPhone or iPad, contact Apple Support.

Let us know what happened.