How I can make camera rotate smoothly followed object?

I have a game where is camera top to down and it is driving game. Now I need that camera follow car smoothly not just parenting. This is my code that works with the position. Problem is when the car turns 45 degrees or 90 degrees and then the screen is not turning. So I need to know how I can make the camera rotate smoothly same direction than the car. But camera should rotate only in the x-axis. The y-axis is always 90 degrees and the z-axis is 0 degrees.

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class SmoothFollow : MonoBehaviour {

    public float interpVelocity;
    public float minDistance;
    public float followDistance;
    public GameObject target;
    public Vector3 offset;
    Vector3 targetPos;

    // Use this for initialization
    void Start()
    {
        targetPos = transform.position;
    }

    // Update is called once per frame
    void FixedUpdate()
    {
        if (target)
        {
            Vector3 posNoZ = transform.position;
            posNoZ.z = target.transform.position.z;

            Vector3 targetDirectionZ = (target.transform.position - posNoZ);

            interpVelocity = targetDirectionZ.magnitude * 5f;

            targetPos = transform.position + (targetDirectionZ.normalized * interpVelocity * Time.deltaTime);

            transform.position = Vector3.Lerp(transform.position, targetPos + offset, 0.25f);


            Vector3 posNoX = transform.position;
            posNoX.x = target.transform.position.x;

            Vector3 targetDirectionX = (target.transform.position - posNoX);

            interpVelocity = targetDirectionX.magnitude * 5f;

            targetPos = transform.position + (targetDirectionX.normalized * interpVelocity * Time.deltaTime);

            transform.position = Vector3.Lerp(transform.position, targetPos + offset, 0.25f);


            
        }
    }
}

Could try adding in this after your movement script

Vector3 targetRotation = target.transform.rotation.eulerAngles;

Vector3 cameraRotation = transform.rotation.eulerAngles;

transform.rotation.Euler(Vector3.Lerp(cameraRotation,targetRotation));

I see others posted as well by the time I setup the environment and wrote the code, but give this a try:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class SmoothFollow : MonoBehaviour {
	public Transform target;
	public float followSpeed = 3f;
	public float rotationSpeed = 3f;

	float distance;
	Vector3 position;
	Vector3 newPos;
	Quaternion rotation;
	Quaternion newRot;


	// Use this for initialization
	void Start() {
		distance = transform.position.y - target.position.y;
		position = new Vector3(target.position.x, target.position.y + distance, target.position.z);
		rotation = Quaternion.Euler(new Vector3(90, target.rotation.eulerAngles.y, 0f));
	}
	
	void FixedUpdate() {
		if (target) {
			newPos = target.position;
			newPos.y += distance;
			newRot = Quaternion.Euler(new Vector3(90f, target.rotation.eulerAngles.y, 0f));
			position = Vector3.Lerp(position, newPos, followSpeed * Time.deltaTime);
			rotation = Quaternion.Lerp(rotation, newRot, rotationSpeed * Time.deltaTime);
			transform.position = position;
			transform.rotation = rotation;
		}
	}
}

The up and down direction might not be the same axis for you. In that case just change the axis, or let me know what your up direction is and I’ll change it for you.

The position and rotation variables are to save on the performance, so you are not accessing transform.position and transform.rotation that many times.

UPDATE:
Thanks @Tap1 for accepting the answer. Let me know how you get on with shakiness. I did write a quick and dirty target mover script I tested it with on the morning (moved a cube object around) and it seemed smooth to me. Here is that code if it helps:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class CarMover : MonoBehaviour {
	public float speed = 30f;
	public float rotationSpeed = 60f;

	Vector3 rotationEuler;
	float turn;

	void Start() {
		rotationEuler = transform.rotation.eulerAngles;
	}

	void FixedUpdate() {
		float inputHorizontal = Input.GetAxis("Horizontal");
		float inputVertical = Input.GetAxis("Vertical");

		rotationEuler.y += inputHorizontal * rotationSpeed * Time.deltaTime;

		transform.rotation = Quaternion.Euler(rotationEuler);
		transform.localPosition += transform.forward * inputVertical * speed * Time.deltaTime;
	}
}

Also made a video 1 about running it in Unity.