Thank you for your question. According to Apple's own support forum they recommend following these steps.
How to Turn Off Search & Widgets on Lock Screen of iPhone
In iOS 15 or newer, turning off the locked search feature is done as follows:
- Open the “Settings” app on iPhone
- Go to “Face ID & Passcode”
- Scroll down and toggle “Today View and Search” to the OFF position
You may also find it helpful to set your default browser to the Covenant Eyes app if you haven't already done so on the phone. You can learn how to do so by clicking the article link here17.
Thank you again and have a great rest of your day.