SmoothDamp in my FPS controller. Does not feel like it is working.

This is my FPS controller script:

 using UnityEngine;
 using System.Collections;
 
 public class PlayerMovement : MonoBehaviour {
     public float speed = 10.0F;
     public float jumpSpeed = 8.0F;
     public float gravity = 20.0F;
     public float sensitivity = 8.0F;
     public float turnSpeed = 0.02F;
 
     private float moveX;
     private float newmoveX;
     private float DmoveX;
     private float moveXv;
 
     private float UpDownRange = 90.0f;
 
     private float moveY;
     private float newmoveY;
     private float DmoveY;
     private float moveYv;
 
     private Vector3 moveDirection = Vector3.zero;
 
     void Update()
     {
         //Move and Jump
         CharacterController controller = GetComponent<CharacterController>();
         if (controller.isGrounded)
         {
             moveDirection = new Vector3(Input.GetAxis("Horizontal"), 0, Input.GetAxis("Vertical"));
             moveDirection = transform.TransformDirection(moveDirection);
             moveDirection *= speed;
             if (Input.GetButton("Jump"))
                 moveDirection.y = jumpSpeed;
 
         }
         moveDirection.y -= gravity * Time.deltaTime;
         controller.Move(moveDirection * Time.deltaTime);
 
         //Look Left and Right
         moveX = Input.GetAxis("Mouse X") * sensitivity;
         newmoveX = Mathf.SmoothDamp(moveX, DmoveX, ref moveXv, turnSpeed);
         transform.Rotate(0, newmoveX, 0);
 
         //Look Up and Down
         moveY -= Input.GetAxis("Mouse Y") * sensitivity;
         Camera cam = GetComponentInChildren<Camera>();
         moveY = Mathf.Clamp(moveY, -UpDownRange, UpDownRange);
         newmoveY = Mathf.SmoothDamp(moveY, DmoveY, ref moveYv, turnSpeed);
         cam.transform.localRotation = Quaternion.Euler(newmoveY, 0, 0);
     }
 }

When I change turnSpeed value to less than 10 the camera shakes (the closer to 10 the less shaking there is). Whatever I do I don't get the smooth movement that I want.