Iv'e really started to like a girl alot, and she openly tells me that she's bi.

To start it bluntly, me and this girl have been only friends for about 3/4 months now, but whenever we are together, texting, chatting we always openly 'flirt' (we I do) with each other. 
Se has told me that she is bi, and recently had two awful relationships with cheating girlfriends but she is interested in a lad, and told me her type of guy etc, and that she hopes to have a family in the future.
 
When we first became friends I had no incline of what my sexual identity I still don't really know but I'm really attracted to her. When we were getting to know each other she would quickly and asked pretty often personal questions about myself like what my type of girl would be, this confused me as I've never been in a proper relationship before (as I was never fussed until I met her). However I always compliment her, saying how hot she looks, nice hair etc. I always want to touch her in some sort of way, like leaning on her when we watch films privately together. We've even been to the cinema together many times but ourselves, had drinks in bars together, but I don't display the same affection I have for her than I would do in private, but I do flirt with her in public just to a less degree. However she does flirt with me in public and in private such as poking my sides and sitting on my lap, laughing at my jokes, smiling at me, looking straight at me (god the eyes) - i'm confused to whether shes flirting or being friendly.

I really started falling for her once we had a great night out clubbing, and one of my other friends noticed how smiley we both are to each other and how often we look at each other and said I think you two would be great together - it did get me to think a lot more about her and what I really like about her. I want to ask her directly how I feel as I have said to her there's someone I like alot. I've told a couple of other friends that I have strong feelings for her, but made them to make sure they wouldn't say anything. 

And everytime she asks we are always face to face - but I shy away as I'm scared she wouldn't want me. I really like this girl but, I know she has low self esteem due to her past relationships with girls. I just want to be that special person for her but I don't think she will want a relationship with a girl anymore due to the past.

I don't think so. She may have some back thoughts about having a relationship with a girl again, but this can't change her attraction towards you (in case there is such). Also, if you know each other better you must have already shown her you are not the type who cheats and stuff and that she can count on you.
Therefore I think you should go ahead and show her/tell her how much she means to you and that you like her blabla. Use the perfect moment to do it.
Good luck.:)