My new iPhone 13 is stuck in the install now mode

Guessing that you have tried to update iOS and install now is grayed out?

Check

• you are on a good Wi-Fi, (from 15.4 iPhone 7 up, you may be offered cellular update)

• Plenty of spare storage – 6GB at least.

• on charge with more than 50% charged,

• low power mode turned off.

• Fully working cellular network connection via your SIM.

• Check Settings, General to make sure you have no Beta profile installed.

• no VPN installed (Check Settings, General, VPN & Device Management)

Resources:-

Update iphone

Update your iPhone, iPad, or iPod touch - Apple Support

Learn how to update your iPhone via recent Macs here:-->

Update your iPhone, iPad or iPod touch with the Finder – Apple Support (UK)

or older Macs and PC here:-->

Update your iPhone, iPad or iPod touch with iTunes – Apple Support (UK)

KB Article - Iphone will not update

If your iPhone, iPad, or iPod touch won't update - Apple Support