How could I purify my relationship and marry a girl I have had sex with?

You are responsible for having had sex with her, and made that decision... so you just do it.

What do you mean by "purifying" your relationship?

What culture or religion calls for 'purifying' a relationship?

Keep in mind that a large amount of these type of relationship questions come from India. We are ill prepared to answer them since the culture is so different.