In the early 60's more than a handful of black musicians spent all their time in Europe because the audiences were bigger, their music was revered as 'sophisticated, their fan-bases were wealthier, and the studios were hungrier.
Miles Davis, Sonny Boy Williamson, Sun Ra, Howlin' Wolf, etc.
Not only those, but Chet Baker (a white guy who mostly lived and gigged in Europe, but not always), Cat Anderson, a black high note trumpeter who played with Duke Ellington for years, Stan Getz, another white tenor saxophonist, many, many others.
I'd posit it wasn't the segregation that many of these guys and gals experienced in the earlier decades (20s, 30s, 40s, 50s, 60s - less so), it was primarily the ACCEPTANCE of their music in Europe. Let's face it, most of the money in American music has been and always will be in the pop genre, especially in the 80s where Tina made her money. Later in the 90s, rap and hip-hop seems to be where a huge chunk of the money is too. Jazz and blues artists, whom you cite, struggled to make a living except in Europe where they, oddly enough, were more popular and more likely to earn a decent living gigging. Perhaps less racism too, though I don't believe that was too much of a factor beyond the mid-60s.
That doesn't explain why Tina Turner chose to first, live in Switzerland, and second, become a Swiss citizen. She was at the top of her game (and money) when she made the move. Something happened to her that drove her toward making that decision. Was in financial? Hard to say. It certainly can't be said that her music was more popular in Europe than in America -- which is my point. She was a rock star everywhere.