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I don't really know how you handled the movement but it seems that you used something that will generate an acceleration, giving a feeling that we are walking on ice, or something like that. 

Instead I would recomand to do something in FixedUpdate like transform.position += Vector3.right*Time.deltaTime*speed where speed is a float variable 

I recommand you to check the video of BlackThornProd about characters movements in 2d games, it was a huge help to me to get how to implement movements and jumps in a platformer game