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?
 A: _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

A: NO
It means that the transformed distribution is normal. Depending on the transformation, it might suggest a lack of normality of the original distribution. For instance, if a log-transformed distribution is normal, then the original distribution was log-normal, which certainly is not normal.
A: In general, the answer is no. It will be normal only if it was generated by the back transformation that corresponds to your (series of) transformations (see edit below). Nevertheless, there are good chances that the distribution of the transformed data is approximately normal, but keep in mind that not every bell-shaped distribution is a normal distribution. You need more than eye bowling before you start your analyses.
Edit regarding the first sentence in my answer: the transformation must be monotonic. For example, if you take data that was generated by a normal distribution, square it and then apply square root - you will not end with a normal distribution.
