Paul's primary audience was non-Jewish members of the Galatian congregations who were not proselytes but who had come to genuine faith in Yeshua and had been received into the community on the basis of Paul's teaching. They are clearly not circumcised (5:2-3) and had come out of paganism (idolatry) into the community of Israel (4:8). While Paul expects that all will hear the words of this epistle (Jew and non-Jew alike), and he writes with this in mind, his primary audience is the non-Jewish membership, and it is to them that the bulk of exhortations are directed.
The backgrounds to this epistle, proposed by Tim Hegg, yield the following scenario, and thus the purpose for Paul's writing it:
Paul's epistle to the Galatians is often used to argue that the Apostle had come to a new realization about his former belief in Judaism.
Tim Hegg challenges this view in his commentary on the book of Galatians. Tim approaches this text with a fresh understanding that Paul did not give up his heritage but instead saw Yeshua as the fulfillment of what he had longed for. In this framework, Tim sees Galatians not as a work against the laws of God but as a letter that affirms all covenant members, whether Jew or Gentile, are welcomed into the family of God.