I searched in vain for studies using the same methodology as Norton and Summers. But public polling, which is much more readily available, suggests white Americans currently do not say they face more bias than their black peers.
A 2017 NPR poll found that while 55% of whites say that anti-white discrimination is real, 84% say that discrimination against racial and ethnic minorities is real. They didn't ask them to compare the two, but I doubt anti-white racism would win when half of whites can't even bring themselves to say it exist.
A 2019 Pew study asked respondents "How does each of the
following affect people’s ability to get ahead in our country these
days?" 56% of whites said that being white helps a lot or a little,
compared to 14% who said it hurts a lot or a little. When asked the
same question about being black, only 20% of whites said it helps a
lot or a little, compared to 55% who said it hurts a lot or a little.
A 2020 Ipsos poll found that 37% of whites said that being white gives
them an advantage in America, compared to only 5% who said
disadvantage. It also found a plurality of white Americans, 49%-43%,
said that "White people have an advantage, compared to people of
color, in our society."
A 2020 NBC/WSJ poll found that 23% of whites said that white
Americans receive too much special treatment while 17% say white
Americans are discriminated against (58% just think they're treated
fairly). By comparison, 52% of whites thinks black Americans are
discriminated against and 13% think they receive too much special
One explanation here may be the shift in presidency, as you suggest. But in a 2016 Pew poll taken before Trump won the election, white respondents were asked whether blacks or whites were treated more fairly in a host of situations, including in the workplace, in the courts, when voting, when applying for loans, etc. Between 20-50% said blacks were treated less fairly, 40-75% said they were treated equally, but those who said whites were treated less fairly never cracked 5%.
I'd suggest that the real outlier here is Norton and Summers. They never directly asked respondents whether "Whites are more victimized by racial bias than Blacks are," that was just a widely repeated nugget when they were researching another question. What Norton and Summers sought to prove was whether or not white Americans viewed racism as a zero-sum game, i.e., do white Americans believe that declining bias against black necessarily comes at the cost of increased bias against whites. Accordingly, they first asked respondents to ‘‘indicate how much you think Blacks were/are the victims of discrimination in the United States" for each of the decades from 1950's to 2000's on a scale from 1 to 10. Then, they asked the same question about whites being the victim of discrimination.
They argued yes, whites Americans do view racism as zero-sum. Both white and black Americans tend to believe that anti-black bias has steadily declined across the decades, but white Americans believe the decrease was more dramatic than black Americans do, and that it came with a upswing in anti-white bias.
So as you can see, yes, white Americans ended up rating anti-white in the 2000's at around a 4.75 and anti-black bias in the 2000's around 3.5. But the first thing to keep in mind is they were intentionally priming their respondents. They were interested finding out what white Americans would say about anti-white bias after you'd already got them thinking about declines in anti-black bias. In a typical poll, they'd randomize the question order precisely to avoid this kind of bias.
Second, it's not necessarily the case that "the average white American" would rank it this way. Norton and Summers noted that a loud minority of their white respondents, 11%, ranked anti-white bias a 10 (compared to only 2% who said 1). To see why that could skew the results, imagine a data set where 11 white Americans rate the biases 10 and 1, but 89 others rate anti-white bias a 4 and anti-black bias a 5. If you averaged those two groups together, you'd conclude that "on average" white Americans view anti-white bias as worse, rating it 4.66 vs. 4.56. But the reality would be that the vast majority of white Americans don't feel that way. That's an extreme example, but you get the idea.
But the bigger problem with drawing larger conclusions from the study is that the way it was framed, respondents may not have been rating anti-white and anti-black bias relative to each other, but relative to the level of bias in earlier decades. If you asked someone to rate anti-black bias today and then immediately anti-white bias today, you could be pretty sure you have a good idea of which one he views as the bigger issue. But we can't be as confident that if you pulled aside a respondent who said anti-black racism today relative to 1950 was a 4, and then six questions later said anti-white racism relative to 1950 was a 5, he would actually say that blacks face less racism today than whites.
To see what I mean, suppose you're a person who believes that over the years, X has declined and Y has increased a steady rate. How would you answer the questions? The natural instinct would be to say that X was at a 10 in the 50's, a 9 in the 60's, and so on, 8, 7, 6, and finally 5 in the 2000's. And then again, 1 for Y in 50's, then, 2, 3, 4, 5, and then 6 in the 2000's. Just following the natural inertia of the line of questioning, you end up ranking one lower than the other. But at no point did you ever intend to the make direct comparisons between X and Y. ALL you wanted to express is "X has gone down, and Y has gone up."
In other words, a problem here is they've asked respondents to track relative change over time, but there's only ten discreet rungs on the ladder, six opportunities to move up or down, and two people moving towards each other. They've set up a system where a white respondent might end up rating white discrimination higher today higher than black discrimination, not because he believes that, but because he believes other, commonly-held beliefs. Such as:
the man at the top of the ladder never stopped moving, i.e. civil
rights progress in America never stagnated from one decade to the
next. (Indeed, there were be a strong inclination from many to answer that he moved
two rungs in the 60's)
the man at top didn't start at the top rung and the man at the
bottom didn't start at the bottom, i.e. not every white American was
racist in 1950, and anti-white bias existed in some form even back then
the man on the bottom has moved, i.e. anti-white
racism has increased, even if it's not nearly as much as anti-black racism has fallen.
Working from these preconceptions gets at minimum the top man at the 4th rung and the bottom man at the 3rd. From there, it doesn't take much of a push.
I am not a Harvard Business or Tufts professor, but I would say Norton and Sommers very aptly demonstrated how widely held those three views are among white Americans (and not at all widely held among black Americans), but it was a bit of stretch on their part to then compare the final two numbers and conclude "the average White American now believes that Whites are more victimized by racial bias than Blacks are." There's way too many asterisks on that finding.
So I guess to get to the root of the matter, I'm not sure I buy into the premise of the question. I'm skeptical that in 2011, the average white American believed there was more anti-white bias than anti-black bias in the US. If that was the case, yes, it looks like there's evidence public opinion has shifted. But it may be that the study you're relying on is an outlier due to its methodology.