Is it normal that i can't talk to this girl?

So in my university, there is this girl that I've liked for several years but I've never been able to have a conversation with her due to my nervousness and social anxiety. I'm friends with her best friend and she is also friends with my best friend, both of which do very little to help me other than give me advice than to step in and create an opportunity to speak to her. Over a year ago, there was a slight fopa on my part where for a psych class, I created a fake Facebook page to see how many people would accept a friend request from a stranger. She accepted it, and I did wrong by actually messaging her as this alias. She found out it was me and so ensued anger. I thought she hated me for the longest time, but about a month ago, her friend told me that I came up in conversation and she said she never hated me and she was just mad for a while and got over it. I've been trying so hard to approach her and try to muster up a random conversation, but I really think I should lead with an overdue apology. Although I do believe it is just social anxiety, is it normal that I have such an impossible difficulty to approach her and just talk to her?