Well, I finally got to the bottom of the Timeout issues. BibleShared uses SQL Server on the backend to store the bible data, user data, and notes data. Unfortunately, in my haste to get the site up and running, I forgot to properly configure the way it handles logging. This miscalculation finally caught up with me. The log file has been growing at a rate of ~100 MB / day to the point that 10 minutes ago it reached 24 GB.
The server its running on happens to run a set of hard disk drives that backup the data as its being written. When trying to back-up large sets of data there is a noticable delay in writing. The large file size combined with this delay was too much for the web site to handle. It took the delayed response as meaning the database server was down, and it gave up.
Fortunatley, all that was needed was a simple configuration change. It should be up and running smoothly now.
This article provides an excellent technical overview of the problem and the solution: