For the purposes of simplicity, I am going to call the girl I am talking about here my girlfriend, although it must be understood that although we've only gone out twice and she likes me, I do not know if our relationship is at the point where it would be appropriate for me to call her my girlfriend.
Anyway, the reason I am writing is to ask about how I should handle this girl's shy personality. Specifically, I would like to ask how to make her comfortable with me. The only time I can speak to her in person is when we go out, so most of our conversing is done through texting. She admits that she is an introvert, as am I, but I am making an effort to be more extroverted with her. Sometimes when I send her a text, usually of a more flirtatious nature (nothing sexually aggressive, all very innocent), she ignores it and does not respond. The thing is that she does sometimes respond to it, and if the message is not flirtatious, she is almost certainly guaranteed to respond to it, unless she does not have anything to add, at which point the conversation will end until I pick it up again.
Another thing that I find difficult is that I always start the conversation. I know she is an introvert, but she is too shy to try.
Although flirting through text has not worked too well for me, on our first date we were in a pet store and she was holding a guinea pig on her shoulder. Her hair was caught in it, so I brushed it out of her face and continued to play with it a little. I also picked up one of the guinea pig's legs and pulled her hoodie back up, which the guinea pig's slipping leg was pulling down. On the second date, I briefly held her hand to look at a burn on it.
In both instances, she did not retract from me. Also, when I am with her in person, she seems much more comfortable than she is through texting. That is the only reason that I have not given up. Is there anything I can do, or do I just need to be patient?