Antialiasing (or actually the lack of it) causes it. With these vector lines - that are supposed to be perfectly straight even shown diagonally - the problem is that the pixels on your monitor run in a vertical/horizontal network. Now when these diagonal lines displayed (and there's not enough antialiasing on), the line becomes jagged (like steps) and the back face bleeds through in between these steps.
To make the "painting of back faces" simpler, after texturing your front faces (individully, as you need), group the whole thing and apply the same texture on the group as a whole. It will texture everything (including back faces) whose faces are not textured in the previous step (those will be kept intact).
If your video card supports OpenGL (and Antialiasing) nicely, you can set it in Window (Sketchup on Mac) > Preferences > OpenGL > Harware acceleration; if you have it checked and select that option, in a list you will see antialiasing options like x2, x4 etc. The higher you set it, the smoother your lines will be (and the less the back faces will bleed through) although the higher it is set, the slower the performance of your SU can be.
If it doesn't really bother you during modeling, you can keep it low. During export, you can set antialiasing separately if you wish and it won't be so striking in an exported image.