As title suggests, I went to normalize some vocal clips in a song I`m working on and it really messed up. Did this video test to show what I mean.
 
Steps to reproduce: https://www.youtube.com/watch?v=f5tQxFQdXF4
 
*EDIT: (for reproduction)
For this test I simply opened a blank project, inserted an audio track then recorded some test vocals. *Crop recorded audio (not split just crop) after cropping, split them into clips, did some fades (although it makes no difference if you do that or not) selected them all to normalize and as you can see the results are in the video. 
 

Yeh, that was pretty durn weird. I was unable to duplicate that behavior here, though.

Let's try a diagnostic. Make a copy of the project if it contains material you don't want to lose. Please bounce each clip to itself, try to normalize again, and report back with what happens. 

If you bounce the clips after splitting, but still keep them as separate clips its works as expected. It seems to be when you just split them, then normalize. Odd. 

This was reported here in 2016-  http://forum.cakewalk.com/FindPost/3512358
Seems to be the same problem.

Not just me then! :-)

Rasure
If you bounce the clips after splitting, but still keep them as separate clips its works as expected. It seems to be when you just split them, then normalize. Odd. 

The clip is just a graphic representation of a piece of audio, not the audio itself. When you normalize a clip, it's pointing to the entire audio file containing that clip. So clips may be normalized to audio that you can't see. Bouncing the clip to itself makes it independent of the file from which it came.

Yeah I realise that, but it shouldn't mess with the audio, it randomly changes the order of the actual audio clips, as demonstrated in the video, not good. The only workaround is to bounce to individual clips after splitting, time consuming and unnecessary IMO.
 
EDIT: Seems to happen on audio files that are cropped, if you do it on a file that`s directly imported, split in to sections then normalize, its fine. If you imported audio file, crop, then split in to sections and normalize, that`s where the issues happen.
 
Steps to reproduce: https://www.youtube.com/watch?v=f5tQxFQdXF4 

I tested this by importing a 1000 Hz mono sine with a peak of -6 dB, and then duplicating it twice. I used Process -> Gain to lower the first clone to -7 dB, the second clone to -8 dB. I then selected all three and used Process -> Normalize to set them all to -3 dB, and then verified that they all do in fact now peak at -3 dB.
 
I tried selecting them three ways: selecting the track, ctl-click on each clip, and dragging a selection box around them. How they were selected made no difference.
 
I also tried normalizing to a lower peak, -10 dB, and that also worked as expected: all three clips peaked at -10 dB.
 
I then split the clips to see if that made a difference, as described above. It didn't.
 
I've run out of ways to test this. Whatever's going on here, it has to be data-specific. Want to send me a test project so I can try it on the same data?

Thanks Bitflipper, uploaded new video for steps to reproduce https://www.youtube.com/watch?v=f5tQxFQdXF4 happens when you crop the clip after import then split and normalize. No issues when no cropping is involved on imported clips :-)