Girl behaving weird after I approached her, might have turned her off, any help?

There's this girl at my gym who I have been seeing for around a few months. Over the course of time, we exchanged a few looks and stuff, but I didn't think into it.

She was walking somewhere, and she looked at me and smiled before I looked at her, which could have been accidental. I looked down quickly to not get any ideas. After this incident, I've caught her staring at me while I was busy with something, and when i caught her, she'd always quickly look down. She'll also look at me through the mirror when I'm unaware, look at me from a distance, and ignore me when close.
I think she was interested, but me ignoring her smile might have made her back off.

After that we exchanged a few looks. She always looked at me before looking at her, and she held eye contact, but I was always the one to break it first. I have the feeling that she wanted me to catch her looking, there's this lingering eye contact before she looks away.
I asked my friend to help me out. He went up to her and said "Excuse me, this guy wants to talk to you". She smiled and said hi, and we shook hands. I asked her name, to which she replied with it, and then we stood there looking at each other in awkward silence and smiling for 3-4 seconds.

I said "Okay, thanks" and went away for a few seconds. I went up to her again and asked her if she didn't mind that I'm talking to her, she said no. I then asked her if we could talk for 2 minutes, she asked me what I wanted to talk about. I got nervous , said "Never mind", and left.

The next 2 days, I think that she ignored me, so I didn't talk. Then the next 2-3 days she started to look again, but now she'll only do it when I'm not aware,, like looking through the corner of her eye. I also had to talk to her again, which was unrelated to our talk, and she looked at me with a blank face before moving. I think she wanted me to talk again or something. Now she'll always look down when walking past me, and I do the same.

I'm not sure what's going on, any help?
11 mo
Forgot to mention, dating isn't done here, so many girls and guys have no idea about this stuff.
Girl behaving weird after I approached her, might have turned her off, any help?
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