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?

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.
