How to reduce database size?
So two primary things can contribute to database bloat:


If you have versioning turned on for your Sections, it will store every change in the db. The Delete Entry Versions plugin can help with this
If your backup isn't dropping temporary cache tables like templatecaches, templatecachecriteria, etc. that can contribute to the bloat as well


Given that you're seeing the backups get so much larger in a one week period makes me think it's probably the temporary cache tables causing the bloat, unless you're doing an awful lot of editing.

It's unclear how you are doing your backups, but check out the Mitigating Disaster via Website Backups; the Craft-Scripts used therein will not include the various temporary tables in the database backups.

It's also possible that too many cache tables are being created due to the way you're using the {% cache %} tags on the site; check out The Craft {% cache %} Tag In-Depth for details on what might be happening.

An easy way to check what's going on would be to connect to the database with something like Sequel Pro or what have you, and look at the number and size of the db rows in the various tables.