When using a web browser, what should a user do to prevent private data from being stored locally?
Browsers can be configured to use a private mode that does not store any data locally.
Set up a second profile and use only that one to access sensitive sites.
Delete the profile directory in the home directory and create a new one after the work is done.
Use the secure versions of the browser available.
There is no way to achieve that because Linux always logs network data.