Broken needles happen for a variety of reasons, in your case I doubt that it is too small. Needles can break for lots or reasons, like it's hitting the bunched up threads or getting caught up in them. It could have a burr that you don't even see, and it could be defective.
Is the thread balling up under the needle plate?
If so, the machine is not threaded correctly, or the top tension is too LOW.
Whether it is the thread, the needle, or the layers is only a matter of figuring out which. Tension is simply a balancing act between these factors. So the first thing. Make absolutely sure the machine is threaded correctly. That is an easy thing to mistake. Get out the manual and double check. Find a needle that fits your thread, make sure the needle is inserted in the correct direction and that the thread path through it is correct. For instance on my Bernina the thread goes through the needle from the front, on very old Singer I have the thread comes from the back through the eye of the needle. Once you are certain that the machine is threaded correctly, then test stitch on some other fabric. If that works, then you know it is not your machine. If there is still looping or birdnesting then the tension on top is too loose, adjust the tension and keep testing. Once everything looks good, try the layers most likely with the walking foot on for best results.
Let us know what happens.