How do I make a vehicle turn properly?

When I tried using a movement script my car turns 90 degrees instantly instead of slowly turning like a real car would.

Here is the code Im using for it

 using UnityEngine;
 using System.Collections;
 
 public class PlayerMovement : MonoBehaviour {
 
 [SerializeField] private Transform target; //created to attached object to
 
 public float rotSpeed = 1.0f;
 public float moveSpeed = 250.0f;
 public float jumpSpeed = 275.0f;
 public float gravity = -9.8f;
 public float terminalVel = -300.0f;
 public float minFallSpeed = -1.0f;
 
 private float verticalSpeed;
 private CharacterController _controller;
 private ControllerColliderHit contactCollider; //used to store raycast information
 
 private PlayerManager playerInfo; //creates a player profile to edit and add info
 private Controller playerController; //recieve inputs from a controller
 
 void Start()
 {
     verticalSpeed = minFallSpeed;
     _controller = GetComponent<CharacterController> ();
     playerInfo =  GameObject.FindGameObjectWithTag("GameManager").GetComponent<PlayerManager> (); //find tag to access scripts from other gameobjects
 }
 
 // Update is called once per frame
 void Update ()
 {
     //start @ 0,0,0 and add movement components progressively
     Vector3 movement = Vector3.zero;
 
     float horInput = Input.GetAxisRaw ("Horizontal");
     float verInput = Input.GetAxisRaw ("Vertical");
 
     if(horInput != 0 || verInput != 0)
     {
         //move while arrow keys are pressed
         movement.x = horInput; //change movespeed to playerInfo.currentSpeed
         movement.z = verInput; //how fast will the player move in that direction while holding the keys
         movement = Vector3.ClampMagnitude (movement, playerInfo.topSpeed); //limit diagonal movement   
 
         //create new Quat to temporarily store the new angle
         //print(transform.localEulerAngles.y); //angle that moves when player is turning
 
         Quaternion temp = target.rotation;
         target.eulerAngles = new Vector3 (0, target.eulerAngles.y, 0);
         movement = target.TransformDirection (movement);
         target.rotation = temp;  //save to replace original rotation
 
 
         //calculates a quat facing in that direction
         //transform.rotation = Quaternion.LookRotation (movement);
         Quaternion direction = Quaternion.LookRotation(movement);
         transform.rotation = Quaternion.Lerp (transform.rotation, direction, rotSpeed * Time.deltaTime);
     }
 
     //if I hold down space
     if(Input.GetKey(KeyCode.Space) || Input.GetKey(KeyCode.Joystick1Button0) )
     {
         brake ();
     }
 
     //if I let go of the brake button
     if(!Input.GetKey(KeyCode.Space) || !Input.GetKey(KeyCode.Joystick1Button0))
     {
         //movement speed goes first to prevent speed from stopping at 1 due to update function
         movementSpeed (); //increases speed when not braking
         playerInfo.hasStopped = false;
     }
 
     movement *= Time.deltaTime;
     _controller.Move (movement);
 
     //move forward
     movement.z = playerInfo.currentSpeed;
     movement *= Time.deltaTime;
     transform.Translate(movement);
 
 }
 
 void OnControllerCollidorHit(ControllerColliderHit hit)
 {
     contactCollider = hit;
 }
 
 
 
 void movementSpeed()
 {
 
     //player hasnt reached max speed
     if(playerInfo.reachedMax != true && playerInfo.hasStopped != true)
     {
         playerInfo.currentSpeed += 1; //if not accelerate
     }
 
     //if at maxSpeed stop increasing
     if(playerInfo.currentSpeed == playerInfo.topSpeed)
     {
         playerInfo.reachedMax = true;
     }
 
     //checks to change reachedMax incase of item pickup
     else
     {
         playerInfo.reachedMax = false; 
         playerInfo.hasStopped = true;
     }
 }
 
 //lower speed until 0
 void brake()
 {
     //while slowing down to 0
     if(playerInfo.currentSpeed >= 0)
     {
         //lower by 10 every frame?
         playerInfo.currentSpeed -= 10;
         playerInfo.hasStopped = false;
     }
 
     //keep current speed at 0 to prevent negative speed
     if(playerInfo.currentSpeed <= 0)
     {
         playerInfo.currentSpeed = 0;
         playerInfo.hasStopped = true;
     }
 
 }
 
 
 //doesnt revert speed back to current
 void boost() //boost boost boost boost
 {
     //create temp to revert boost back
     float temp = playerInfo.currentSpeed;
 
     if(Input.GetKeyDown(KeyCode.LeftShift))
     {
         playerInfo.currentSpeed *= 1.5f; //since its private it cant change values - going to public set should fix
     }
 
     if(Input.GetKeyUp(KeyCode.LeftShift))
     {
         playerInfo.currentSpeed = temp;
     }
 }

}