In the US, Jews began to become highly represented (although 'dominate' is inaccurate as this would imply they make up the largest ethnic group in these fields overwhelmingly, which they don't) in fields like media and law once overt anti-Semitism and 'clannishness' among the WASP population declined from the 1930s onwards. This period also saw the rise of other non-WASP groups, such as the Irish and Southern and Eastern European Christians.

As for why Jews were able to take advantage of these new opportunities so especially effectively, it's clear that there was a high level of ability and drive to succeed in these fields among the Jewish population.

One cultural explanation for this is the Jewish focus on education and business caused by their long history of being forced into financial occupations due to their religion in Europe. Another is that they had drive due to being a small and insecure minority in a larger culture. See Amy Chua's 'The Triple Package' for an explanation of this one.

Some thinkers (e.g. Cochran and Harpending) have claimed that the focus on education and business in Jewish culture due to long experience of occupation in these fields gave rise to an instance of gene-culture evolution, where there was higher selective pressure among European Jews for high mental ability than in the surrounding populations.