If a data set appears to be normal after some transformation is applied, is it really normal? Suppose you have a data set that doesn't appear to be normal when its distribution is first plotted (e.g., it's qqplot is curved). If after some kind of transformation is applied (e.g., log, square root, etc.) it seems to follow normality (e.g., qqplot is more straight), does that mean that the dataset was actually normal in the first place and just needed to be transformed properly, or is that an incorrect assumption to make?

_Comment continued: Consider lognormal data x, which does become
exactly normal when transformed by taking logs. In this case (with $n=1000),$
Q-Q plots and the Shapiro-Wilk normality test agree for original and transformed data.
set.seed(2022)
x = rlnorm(100, 50, 7)
y = log(x)
par(mfrow = c(1,2))
 hdr1 = "Lognormal Sample: Norm Q-Q Plot"
 qqnorm(x, main=hdr1)
  abline(a=mean(x),  b=sd(x), col="blue")
 hdr2 = "Normal Sample: Norm Q-Q Plot"
 qqnorm(y, main=hdr2) 
  abline(a=mean(y), b=sd(y), col="blue")
par(mfrow = c(1,1))


shapiro.test(x)

        Shapiro-Wilk normality test

data:  x
W = 0.1143, p-value < 2.2e-16     # Normality strongly rejected

shapiro.test(y)

        Shapiro-Wilk normality test

data:  y
W = 0.99017, p-value = 0.678     # Does not rejece null hyp: normal

