A dark-skinned Jesus would at least make some sense.
Those pictures of an overtly caucasian Jesus? Completely fictional, with no basis in historical fact.
Since we were colonized by Europeans, we've been handed down the caucasian Jesus of the Spaniards. Jesus would never have looked like that if he were a pure-blooded Jew (descended from David) as claimed. He'd likely have the trademark Jewish nose (large, not thin with a high bridge), dark skin (more likely Arabic in color) and a swarthy face.
Something like this:
Not quite as popular an image as this:
But a hell of a lot more accurate.