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?

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.
