Am i normal? what is normal?

Well i'll start from the beginning. Im a quiet introverted girl, when I was young I never knew what was wrong with me, I cant mix with people well etc.
I grew up not knowing myself and not making any long lasting friends, never dreamt of doing half the stuff 'normal people' do like drinking and clubbing. I had low confidence and never went out much, I watched everyone grow up, get boyfriends and party and I thought why not me? because I was introverted I hated the thought. But I grew up hating myself, I tried to push myself to do new things and eventually I got more confident in talking to strangers, my cousin invited me round town one night and of I went, I now go out whenever she does which isn't often but often enough, finally I felt normal.
But I still don't have friends to meet up often with or to call my own, I have my cousins which are all so nice. I felt like I was letting my parents down, I had never had a boyfriend and it made me low, now 23 and nothing. My mum pushed me to go out with someone else and it killed me because I didn't fancy him.
I met someone on a random website and became good friends, I f ell for him and I couldn't tell my parents as I was so scared they don't get the whole online thing. Until my mum asked me to try it so I told her about him she was supportive and said icould meet him and asked about him, he cant meet until later this year, and I actually told him no in the first place because of parents. But now my mum is okay with it I said yes.
My mum has now come back after 2 weeks and saying I need to try online dating sites, I said I wanted to meet the lad I really like... and she backtracked again.. saying I don't know him and hes too far and how long have you been waiting.
We've been talking 2 years only really spoke of more for about the second 11months. And I said no at first so ive been waiting 2 weeks actually.
She now wont talk to me because I wont try a dating site. Ive looked before and I just cant do it, I know who I want to meet and if it doesn't work I will try alternatives, but shes dead against it and I hate it because I feel so alone again :(
Is it normal behaviour? should I be looking everywhere?
If the lad I know online isn't safe then surely randomers off dating sites who I haven't spoke to and probably wont for the length I have with him aren't??