Hey Sonoroman,
Thanks for using Apple Support Communities! I understand since upgrade the iOS on your iPhone, the Screen Recording isn't working as expected. This is a great feature, and I definitely want to make sure you're able to use it without issue.
If you haven't already done so, let's test if this behavior continues after restarting your device. Use the steps to restart here: Restart your iPhone, iPad, or iPod touch
You may also consider updating the iOS on your device to the latest version of iOS 13. You can see how to do that here: Update your iPhone, iPad, or iPod touch
I would also like to get a bit more information so we can continue working toward a resolution. Can you please answer the following?
Does this behavior happen with in all apps, or only some?
When Screen Recording does stop, is there an error? If so, what's that error say?
Does this behavior happen on both Wi-Fi and cellular data?
Thanks again, and have a great day!