I have had the same experience where The wood broke, for what ever the reason. When I first started using the scroll saw, my family penned the nick name "Tooth pick maker". It was a disaster. That was when I really began to utilize different tecgniques and search for other peoples advice. I even stopped purchaseing lumber from the local lumber yard.
I use several tricks that seem to help. First, I stack cut, using 2 pieces that I put box tape over the wood with a small amout tapered over the edge to hold the wood together, and secure the pattern to the tape. The bottom board gives the upper one added strength. I also cut the small areas first using either a #3 or #5 skip tooth blade or which ever blade I find serves the need. I do not use spiral blades because I have a hard time contorlling where they cut and they have a tendency to splinter the wood (personal preference). I have found that when the larger areas are cut it has a tendency to weaken the wood. When drilling the holes, I use a tiny wire bit and place scrap wood beneath to prevent splintering. As for the long thin lines, Generally I cut down one side back the blade out, and then cut down the second side. I do this because sometimes in turning the wood, the blade thickness results in straying off the path. There are times when I am cutting that I find the fibration of the area being removed is vibrating a lot, and I use scotch tape over the cut areas to reduce this. I do so because I have found that the more the wood vibrates, the gerter the opportunity for it to break a weak wall.
As grandpa shared, do not push the wood, let the blade do the work. The more one forces the wood the odds of breakage of lumber and blades. Yes, even with these tricks, things brake and I have to glue them.
The real trick is to be patient with yourself and the learning experience. The more I try to cut things perfectly, the more stress, and the more strees, the more frustration. Just have fun with it. Try an easier project for a while and come back to the portrait. It will still be there.
My two sense!