How to delete browser history in safari?

If the option to clear your browser history is grayed out, it's either because you've already cleared it, or you don't have permission and your device is restricted by either restrictions/screen time or a management profile if you use it for work.

https://support.apple.com/en-us/HT201304

First thing I would try is to restart and/or force restart your iPhone:

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

If your iPhone, iPad, or iPod touch won't turn on or is frozen - Apple Support

Then see if you can clear the browsing history.