Note: I am a white transwoman, and I have not had to have any of the experiences of bleaching or similar, but I have tried to read up on the subject a bit to be informed more on this subject, but obviously, I am still going to be lacking. I apologize if anything here is incorrect. Please let me know and I will reword or rewrite.

Also content warning: this post discusses sexual assault, abuse, and slavery.

How "widespread" it was is not really known, from my investigation, but it was definitely a phenomenon that existed. Skin bleaching was advertised in 1920s Harlem as a way for people of color to integrate and was seen as needed to get better jobs and have better outcomes in society (see Dorman 2011).

It particularly rose in popularity in the early twentieth century in urban Southern cities and among black women specifically, because it was engrained into the conceptualizations of social beauty and acceptableness and attractiveness, and therefore white skin as beauty became a concept that was grasped upon (Lindsey 2011). A major cause of this was the domination of white males heading the beauty industry. Lindsey writes:

From its inception, African American beauty culture was simultaneously syncretic and adaptive. White manufacturers, which dominated the beauty culture industry, however, relied on white cultural ideals in product development and marketing throughout the nineteenth century. These manufacturers used prevailing white aesthetic tastes to target African American men and women with the ability to purchase beauty products. Both enslaved and free black women confronted white cultural dominance and beauty norms that degraded and dehumanized black women. White Americans and eventually, some African Americans perceived the features of enslaved and free black women as physically unattractive and as indicative of the “inherent” primal, animalistic, and lascivious tendencies of peoples of African descent. (Lindsey 2011: 101)

Also, it should be kept in mind the pre and post-slavery contexts, where the sexual characteristics of black women was abused, exploited, and degraded by white slave owners, and then used as a way to discriminate against them. Black women were sexually abused horrifically under slave owners (if you want a picture of slavery, Twelve Years a Slave by Solomon Northup is suggested highly), and afterwards then demonized and stereotyped mercilessly (and many of those stereotypes continue today; and then many more have been replaced, specifically with the fetishization and stereotyping of non-white bodies by the white dominated beauty industry). The results were that skin bleaching became increasingly popular in some areas. This was prominent still enough in the 1960s that Malcolm X specifically talked about it in one of his particularly illuminating (and in my opinion one of his best) speeches given April 27, 1962. He said (you can see the recording here):

Who taught you to hate the texture of your hair? Who hate you to hate the color of your skin to such extent that you bleach to get like the white man? Who taught you to hate the shape of your nose and the shape of your lips? Who taught you to hate yourself from the top of your head to the soles of your feet? Who taught you to hate your own kind? Who taught you to hate the race that you belong to, so much so that you don't want to be around each other? No before you come asking Mr. Muhammad does he teach hate, you should ask yourself who taught you to hate being what God gave you!

Needless to say, systemic racism built into the beauty industry (which still pervades it to this day), was quite horrific and led to a lot of black women to bleach their skin in order to better conform and be perceived as beautiful at all. Your title is, well, just incorrect. Non-white people most certainly did do this, and there are still cosmetic surgeries and procedures that have been undertaken (to this very day) for people of color to conform to racist expectations in beauty and abroad. A particular one that has become politicized today has to do with black people's hair and hairstyles.

Sources:

Jacob S. Dorman, "Skin Bleach And Civilization: The Racial Formation of Blackness in 1920s Harlem," Journal of Pan-African Studies 4, no. 4 (2011): 47-80

Treva B. Lindsey, "Black No More: Skin Bleaching and the Emergence of New Negro Womanhood Beauty Culture," Journal of Pan-African Studies 4, no. 4 (2011): 97-116

Solomon Northup, Twelve Years a Slave, edited by Sue Eakin and Joseph Logsdon (Baton Rouge: Louisiana State University Press, 1968) is a highly suggested edition.