If I am in the US on a student visa and a US girl wants to marry me, is that possible?

You can marry an American citizen in order to get American residency on a permanent basis. The process is as follows:

If you are a US citizen and are planning to marry a citizen of another country (a foreign national), or are from another country and are planning to marry a US citizen, the procedure to obtain permanent residency in the US for the international spouse is described below. Foreign students and scholars are welcomed and encouraged to consult with ISSO staff about the steps and timing of this procedure.

In our local area, the procedure can take 6 - 12 months from the time the petition is filed until the international spouse becomes a permanent resident of the U.S.. Once he or she becomes a U.S. permanent resident, s/he will continue to travel on his/her home country passport, so it is important that the name on the passport and the final I-551 permanent residency card be identical.

Note: This process is called "adjustment of status to permanent resident," and is not the process for becoming a US citizen; that process is called "naturalization." If you are married to a US citizen, you are eligible to go through the naturalization process after you have been a permanent resident for three years. You are not required to go through the naturalization process; few occupations require US citizenship. However, as a citizen you have the right to vote and to hold public office.