In theory it should, but it often does not. The data will remain on your phone and be continually backed up into iCloud. Further, there is no easy way to go back and delete all URLs. Lastly, you would be losing that data permanently, while using the described backup method, you lose nothing.
On macOS there is a parallel "system data" problem, I searched for the oversized folder, found a bunch of cache files with an unusual extension, found an online converter and realized it was a URL preview from messages.
justachos OP t1_jczevbn wrote
Reply to comment by Newphonewhodis42 in I solved the “system data” storage hogging mystery by justachos
Please tell!