How can I seamlessly loop with turbulent displace filter in AE?
In order to get a fluid and seamless loop, it's key to enable Cycle Evolution and move the last keyframe 1 frame beyond the end of the composition.

Add the Turbulent Displace Effect onto the footage

Open up Evolution Options

Enable Cycle Evolution

Go to the first frame in the timeline

Click on the stop watch of the Evolution value:


Go to the last frame in the timeline and set the Evolution Times value to 1x, this is equal to 360 degree (one revolution)

Move the second keyframe 1 frame beyond the end of the composition:



Example


Composition length: 51 frames
First evolution keyframe value: 0x at frame 0
Second evolution keyframe value: 1x at frame 51

Note: This seems to work with all fractal effects.