White men/black women in the USA was going on even before the fist slaves arrived in that the slave ships' crew often "used" female slaves under transport. In the antebellum south the use of female slaves forsex was one of those things that was obvious - after all where did those lighter skin kids come from - but was never mentioned in polite society. Up through the early 20th century the houses in New Orleans would grade black prostitutes by their known or implied race mixture (mulatto, quadroon, octaroon etc). Institutional racism should not be confused with the ability of the staunchest segregation supporter to show human decency to blacks, although they had to know their place. Overt, nasty, person to person racism was considered to be low class and not done by "better" folks. The point of this is the ability of Thrurmond to have a black woman as a mistress or a casual sex interest, or to go to bat for a black constituent of the "worthy" sort (like a veteran) does indicate some sort of cognitive dissonance, but rather is completely in line with the "old south" attitude on race.
I doubt any white man was ever prosecuted for a covert relationship with a black woman, it was only when it became open or tried to lead to marriage (see Loving v Virginia). In the 40s a white man in the south having a covert relationship with a black woman, even with pregnancy, would be a social severe tut-tutting. A person of stature (or anyone) being outed as gay would spike their career.