That is absolutely not true. One can be in a dominant group, but situation still arise where one is bullied, harassed, or otherwise discriminated against. Furthermore, it is sheer bigotry to assume that someone who is white automatically has privilege -- there are plenty of poor whites from broken homes who have suffered from trauma, and wealthy blacks who live in gated communities.
When my children were teens, we lived in a Barrio, because it was all I could afford. My son was routinely beaten up at high school while called all sorts of vile names for white people. He couldn't walk down the street to his friend's house because of knives and guns being pulled on him, telling him to get out of the neighborhood.
Does that sound privileged to you?