Everything the other attorneys on here have said is absolutely correct, but just to offer some extra explanation:
As an Asylee, you have the ability to file a petition for spouses and children within two years of your grant of asylum through an I-730, but the relationship must exist prior to the grant of asylum (so, your girlfriend/fiance does not qualify for this - just explaining as you may have heard of it and wonder why you cannot use this for your fiance). Since your girlfriend/fiance does not qualify, you would have to petition for her as a green card holder or, eventually, a US Citizen. You cannot petition for a girlfriend/fiance as a green card holder (only US Citizens). You would have to marry her to file a petition for her as your spouse while you are a green card holder, but the wait for a visa would be about 2.5 years; this is NOT when she would come to the USA - this is when you could start the Affidavit of Support and Immigrant Visa Document Processing (so, pay fees, prove you are able to sponsor her or have joint sponsor, submit a bunch of document from her for the visa, etc.) so the file could go to the Embassy and she could go to interview. If everything went well at the interview, she would be approved and then could come to the USA. Whether this would be any faster than waiting till you are a US Citizen and filing for her as a fiance depends on visa availability which is always changing. Be prepared to sponsor her (promising and proving to the government that if she comes to the USA, she will not be a "public charge" and not get public benefits until she earns 40 credits w/Social Security or becomes a US Citizen herself - you prove this by having sufficient income to support your household plus her) or have a joint sponsor.
Now, you can marry her anytime you want, but where and how? It might be very difficult for her to visit you depending on her situation. You are an asylee - you should not return to your country of origin or you may risk your status. If you travel, you should not use your country's passport (always use a Refugee Travel Document or Reentry Permit to travel on while a green card holder).
You need to know these things so you can plan accordingly and make informed decisions. If you want to be prepared, try to consult with an immigration attorney or two about your goals, get advice you can rely on, and then plan accordingly. If money is an issue, try contacting a local area legal aid and/or non-profit legal immigration service provider in your area to get advice on this now so you do not unnecessarily delay your case(s) in the future. Best of luck to you and your girlfriend. I hope this helps and please feel free to ask any follow-up questions and I will do my best to give you some direction.
You can petition for her if you have a green card. If you filed today it would take about 2 1/2 years
Samuel Ouya Maina, Esq. 415.391.6612 email@example.com Law Offices of S. Ouya Maina, PC 332 Pine Street, Suite 707 San Francisco, CA 94104