app downloads - automatically install free and paid apps - will not turn on

Check if this helps -> Use parental controls on your child's iPhone, iPad and iPod touch

Use parental controls on your child's iPhone, iPad and iPod touch – Apple Support (UK)

• Check restrictions -> tap Settings -> Screen Time -> Content & Privacy -> scroll to Allow Changes.

Check other restrictions:

Manage App Store purchases, subscriptions, settings, and restrictions on iPhone

Manage App Store purchases, subscriptions, settings, and restrictions on iPhone – Apple Support (UK)

Prevent in-app purchases from the App Store

Prevent in-app purchases from the App Store – Apple Support (UK)

The button for app downloads stays on for 2 seconds, then turns off...

Weird . . .

Try to do a force restart -> Force restart iPhone – Apple Support (UK)

If it doesn't help, try to check the network connection:

You need to have a good Service and/or Wi-Fi.

If they are On, turn Off Hotspot & VPN.

Charge the battery > 70%.

If you can't connect to the App Store, iTunes Store, or other Apple services

If you can't connect to the App Store, iTunes Store, or other Apple services - Apple Support

-> If you tried all of the above steps and you still can’t connect, contact Apple Support.

I'm the first "me to." It's not your single device. I have five iDevices; none of them will "stick." It turns on for a couple of seconds then turns right back off, as you have said. I guess we'll see what develops.

Get more help and assistant from Apple directly, see Contact Apple for support and service -> Contact Apple for support and service – Apple Support (UK) 

👍 that is good.

It is fixed now.👍🏻