This is one the questions that pops up in my head all the time. I even asked this question on my Facebook page, and I also asked my friends at work. After doing my research on Youtube, Google, Ask.com, Yahoo Answers, and from personal experience, this is what I came up with: NO.
There is only one answer, unless somebody can prove me wrong. Your hair looks like it is growing faster because you are experiencing less breakage. Having too much chemicals on your hair can make it really dry. And if you don't moisturize your hair everyday while it is relaxed, your dry ends will break off, which will make your hair look like it is not growing. Improper application of the relaxer on your hair, overlapping, not rinsing the relaxer completely with neutralizing shampoo, can also damage the hair and breaks it off.
There you have it! If you were thinking about reading all the Google pages about that answer, don't even bother, I have already done all the work for you. Natural hair does not grow faster than relaxed hair.