How to lock a browser with a password/passcode on iPhone?

Solution 1:

iCab Mobile has had this feature since at least 2012.

iCab » Settings » Account

(iCab is, by far, the most feature-rich browser on iOS. I’ve referred to it as “Safari Pro” for many years.)

Solution 2:

There is no built-in means to secure only web browsers on iOS with a passcode that wouldn't also affect the entire device.

Consider erasing the iPhone and enrolling it in an MDM, or mobile device manager, such as Jamf Now.

To proceed, you'll need to use Apple's free Configuration Manager 2.

While you can enroll an iOS device into an MDM (not just Jamf) without 'supervising' it, to have the greatest amount of control over the device you should set it as supervised. Note that you cannot restore an unsupervised iOS backup from iTunes, Configurator or iCloud to a supervised device.

Once supervised, enroll the device into Jamf Now and choose a 'blueprint' of security options, including remotely managing ScreenTime and disabling the installation of other apps.