Couple days ago I went out with this girl and everything went well. By the end of the night we were making out in the car.

Me: What happens now?
Her: We continue seeing each other.

I didn't see her for 4 days because we were both busy and one of us was out of town for school. Today i saw her and didn't' initiate with a kiss because I nervous/unsure where we left off. Day went by and i saw her again in the afternoon.

Her: Hey!
Me: Hey, I came back to apologize.
Her: For what?
Me: I should have done something when I first saw you earlier.
**I went in for a "sensual kiss"**
She enjoyed it

She called me later and said she wanted to talk about us.

Summary of Conversation
Was happy to see me again and enjoyed that moment we had but she wanted to tell me she is shy about showing affection in public because she feels awkward about it. She wanted to tell me after she saw me but she was spaced out after that and was too excited see me. She said she likes it when we're alone. I told her alright and that I was sorry if I made her feel uncomfortable. She was happy i understood.
Then we just talked about random shit for like 15-20 min.

What are my faults?
Advise?

What's the problem? She just doesn't like kissing in public. It makes her nervous. That's a problem with her not you.

The only way to stop this behaviour is to desensitise her to PDAs as in take control and kiss her in public however you'll have to make each time enjoyable for her so she sees reward in it. Think of it as positive reinforcement having her enjoy the kiss in public. Gradually she'll grow accustomed to it and think nothing of it.

Generally what I do is treat random and spontaneous kisses as moments you know the kind of moments women harp on about in romance novels -- spontaneous, care free, exciting. It's all down to you simply being blind to your surroundings and treating the girl as if she's the only person there. She'll get drawn into it and get a high from it.

She said I'm her second real boyfriend so maybe she's unaccustomed to the whole showing affection.
I understand, although it seems like I'll be fighting fire with fire. Doing something that makes her uncomfortable. I guess I have to so we can progress.