How can I determine what is causing the used disk space to slowly increase in Linux?

You can find the recently updated files on your system using something like

find / -type f -newermt '-5 minutes'`

You can combine that with a size display:

find / -type f -newermt '-5 minutes' -exec stat -c '%10s %n' {} + `

My suggestion would be this command:

watch -d "ls -lt /var/log/**/* | head"

watch runs the following command by default every 2 seconds. The -d flag highlights differences after each execution.

ls -lt lists files according to their last modified date (newest ones first), **/* is a glob to find all files recursively.

Lastly, head is used to output only the first 10 lines.