How do you overcome this fear of speaking up
Fear is a biologically normal unless it consumes ones life. To over come fear you have to face the fear it self. This may sound cliché, but in most cases fear is self induced. This is resulting from the harm done to the fearful person by their own brain. The brain sees the situation as being far worse than if what is feared actually came about.
These fears may have stems from emotional abuse in years of early development as do most cases of shyness and social anxiety. So look within and internalize the issues and where they stem from to overcome.
Do you think stage fright is related to this?
Yes I think stage fright is directly correlated.