That result is definitely not normal. I don't have a Brother, so not sure about settings on that machine. It's true that in general, if stitches on the bottom are loose then you would tighten the upper tension. However, you are saying that you tightened the upper tension and still had a problem with the underneath stitches, right?
For machine quilting, I have found the following video extremely helpful:
http://www.youtube.com/watch?v=Q1mRhcquZTM
Although the video is about adjusting tension on a longarm machine, the advice works well for domestic machines too. Basically it means loosening the bobbin tension as much as possible first, then adjusting top tension until you get a balanced stitch.
In short, though, the problem you are describing should not be happening. It is likely that there is either something wrong with the balance of your tensions, or something wrong with your machine. Usually the problem is with the tension settings (both bobbin and top).
Edit: Other things to check are (1) that the needle is an appropriate size and type for quilting and (2) that the bobbin is the correct one for the machine, is not distorted, and has been wound properly. Poorly wound bobbins can cause a myriad of problems.