25 is still young, so I don't think most people would find it too odd that you haven't had a girlfriend yet.
A lot of young men prefer to play the field and date several women or leave their romantic involvements to only one night stands.
Age is in your favor. It would be a whole different story if you were 45 and still living with your mother.
I most definitely would NOT make up a "girlfriend". One lie leads to another and you will get caught. People do not care about your past as much as you do. People will judge you on who you are "now"....and nobody likes a liar. Everyone's had a bum card dealt to them in life and we all have skeletons in our closet.
Have you thought about seeing a therapist to help you over come your social anxiety?