Hi,
The Normalize modifier is not intended to be used with animated hair. In this case, if you still want the overall effect of this modifier, you have to bake the results into an Edit Guides modifier before/when applying animation. See how here:
https://www.youtube.com/watch?v=3FtVS55VFMc
As you found out, the flickering happens because Normalize Strands is recomputed at every frame, so it normalizes the strands every frame, causing indesired movements.
Jeordanis Figuereo (Product Designer. EPHERE Inc.)