A significant number of people who fear flying develop claustrophobia, a fear of being closed in. This can be controlled by learning to breathe from your diaphragm. Hyperventilation and claustrophobia both occur when breathing with a tense upper chest. This will also result in an increase in the heart rate. To change your breathing patterns, practice concentrating on your belly when breathing.
Massage can also help relax tense chest muscles which tends to impair proper breathing.
Aircraft turbulence is frequently a cause for anxiety among travelers. Although at times turbulence can cause discomfort, it's important to remember that it is not unsafe. Planes are built to withstand much more than turbulence. As long as your seat belt is fastened, the aircraft can sustain very rough conditions. Also a point to remember is that most airlines usually don't fly through thunder storms, they fly around them.