Can my teeth be re-aligned so that I would have a normal bite when I move my lower jaw forward? It makes my side profile better.

What you describe has been called "Suck Back Orthodontics" it was commonly done in the past but is much less common today.  You did not mention if you had sleep apnea, snoring , headaches, or ™ Joint problems which are common occurrences to the jaw being pushed too far back.The DNA Appliance can grow your upper jaw larger as an cult to allow the lower jaw to come forward.  It has been shown in case reports to cure sleep apnea and/or snoring.  It will definitely allow you to have the profile with a stronger chin.  There are numerous positive health outcomes that could also occur in conjunction with correcting profile.

For this concern, I would recommend you visit an orthodontist. Orthognathic surgery may be a healthier and safer option for you.