How did it come about that Jesus was portrayed as White in media?
There is an interesting article published by popular mechanics that uses modern forensic science on a galilean skull dating roughly to the time of Christ. (The Real Face Of Jesus). Despite the title of the article, it is not the real face of Jesus, but it makes a good argument as to what people living in the area may have looked liked (Jesus included).
If you could travel the world and visit the churches of different nationalities you will see that Jesus is nearly always portrayed resembling the dominate ethnic group. Here is good article with examples on this topic from Intervarsity / Urbana (Race and Ethnicity in the Bible). Your observation of Jesus as being white is a reflection of a dominate western "white" culture that has painted Jesus according to what is familiar. Positivity - cultures depicting Jesus as ethnically the same are telling the story of how the Grace of God includes that specific ethnic group. Negatively - the depiction of Jesus as ethnically the same as the dominant racial group has lead to forms of prejudice and feelings of exclusion among those who do not identify with the dominate group.