Try this, once you match the seams of the halves, pin them together for joining, and remove the paper where you will be stitching. The paper is to stabalize the units until they are joined, but can cause problems in and of itself. You did nothing wrong; it is impossible to sew through that many layers of paper & fabric. Another thing you might try is pressing some of your seams open where several will come together, as in the center of your block.
Hope this helps. :D
CP