Move character to mouse position, keeping constant speed

I'm writing a character control script for a top-down RPG, and want my character to move towards the mouse position whenever I click or hold the mouse button down:

 using UnityEngine;
 using System.Collections;
 
 public class Movement : MonoBehaviour {
     
     //Player movement speed
     public float movementSpeed = 10.0f;
     
     //Player height
     public float height = 0.5f;
     
     //Target position
     Vector3 targetPosition; 
     
     void Start()
     {
         //Setting target position to current position upon game start
         targetPosition = transform.position;
     }
         
     // Update is called once per frame
     void Update ()
     {
         //When user presses left mouse button
         if(Input.GetMouseButton(0))
         {
             //Update target position
             targetPosition = Camera.main.ScreenToWorldPoint(Input.mousePosition);
         }
         
         //Movement code attempt 1 (obviously I don't use both at the same time, I've merely tried both:
         transform.position = transform.position + (targetPosition - transform.position).normalized * movementSpeed * Time.deltaTime;
         
         //Movement code attempt 2:
         transform.position = Vector3.Lerp(transform.position, targetPosition, (Time.deltaTime*movementSpeed));
     }
 }
 

It works just fine, aside from that my character moves slower when traveling shorter distances, and faster when traveling longer distances. I can understand why this happens, as the movement time is constant no matter how far you travel, but I'm not sure how to fix that - I want the movement speed to be the same no matter what. I've tried both approaches above and in both cases this happens.

I took a look at some other similar questions in here, but found nothing that worked in this case. Any ideas?