Do you by chance have a Windows computer with iCloud Control Panel installed on it?
What I've noticed is if you have iCloud Control Panel installed on Windows 8+ while Windows is syncing your bookmarks with your Microsoft account, it causes this problem.
The reason I've come up with is that iCloud creates bookmarks on Windows, which Windows will in turn upload them to your Microsoft account, which changes them in a way that iCloud doesn't like it so it pushes the changes out to your phone via iCloud, which then changes it to a format it likes, and sends that back to your Windows computer via iCloud and again starts the vicious cycle of battery drain.
Remove iCloud from your computer or turn Safari sync off on your phone and see if this fixes your battery drain issue. If you have more than one battery powered Apple device, you may want to do the same on those.
Alternatively if you use Mac more than Windows and only have 1 Windows computer, you can just turn the bookmarks/favorites sync off on your Windows computer to your Microsoft account.