If you outgrow a particular way of behaving or thinking, you change and become more mature, so that you no longer behave or think in that way. The girl may or may not outgrow her interest in fashion. = grow out of
If a child outgrows a piece of clothing, they grow bigger, so that it no longer fits them. She outgrew her clothes so rapidly that Patsy was always having to buy new ones. = grow out of