Screen time full even when iPhone is on flight mode

Hi there. I am quite a newbie on ios but I could see that sometimes my screen time has been at his maximum (60 minutes per hour) in the previous hours although I was sleeping and phone was on flight mode. It seems that it is due to pages I visited after clicking on a link in an email or another app. I can then set a limit to stop it but I don't think it's the solution.

I have 2 questions:

1. Does it drain the battery even if my phone is locked?
2. Do you know where this is coming from? How can I prevent it?

Thanks in advance for your help.