Tightening up turning radius with AddRelativeForce (or other options)

Hi, how is everyone? I tried asking this in the forums, but I guess no one wanted to help there, so I'm asking it here.

I have been working on this game project for a school studio project (and possibly it might end up being more if I could get this one thing working correctly). The idea is to be a arcade space racing game, subsequently, I want to have physics that are quick (like the ability to turn yaw-wise in a quick fashion).

The issue I keep running into is how to pull this off? I have the script working, I have barrel rolls, strafing, but I am using rigidbody.AddRelativeForce as the means of propulsion/movement. This due to the fact that I couldn't find anyway outside of using Unity's physics system for this. I tried transform.Translate, but that would only propel the ship in one direction, along the z-axis, and not the direction the ship is pointing in. So I am also open to figuring out a non-Unity physics-based method to get this working (if anyone knows, otherwise I'll stick to using rigidbody physics) as the physics system seems laggy/lazy to me turning-wise.

So can anyone help me? Here is my script for just the ship's basic controls.

Thanks for any help.

     using UnityEngine;
     using System.Collections;
      
     public class PlayerShip_Easy : MonoBehaviour
     {
             private float shipSpd = 300.0f;
             private float turnSpd = 150.0f;
             private float rotSpd = 50.0f;
             private float strafeSpd = 50.0f;
      
         float rollRate = 0.0f;
         float pitchRate = 0.0f;
         float yawRate = 0.0f;
      
             float invert = -1.0f;
      
     void Update ()
         {
             float roll = Input.GetAxis("Roll");
             float pitch = Input.GetAxis("Pitch");
             float yaw = Input.GetAxis("Yaw");
             float strafe = Input.GetAxis("Strafe");
      
             if (Input.GetAxis("Pitch") > 0)
             {
                 pitchRate -= pitch * invert * turnSpd * Time.deltaTime;
                 pitchRate = Mathf.Clamp(pitchRate, -45.0f, 45.0f);
                 transform.rotation = Quaternion.Euler(pitchRate, yawRate , 0);
             }
             else if (Input.GetAxis("Pitch") < 0)
             {
                 pitchRate += pitch * turnSpd * Time.deltaTime;
                 pitchRate = Mathf.Clamp(pitchRate, -45.0f, 45.0f);
                 transform.rotation = Quaternion.Euler(pitchRate, yawRate , 0);
             }
      
             if (Input.GetAxis("Yaw") > 0)
             {
                 yawRate += yaw * turnSpd  * Time.deltaTime;
                 yawRate = Mathf.Clamp(yawRate, -60.0f, 60.0f);
                 transform.rotation = Quaternion.Euler(pitchRate, yawRate, 0);
             }
             else if (Input.GetAxis("Yaw") < 0)
             {
                 yawRate += yaw * turnSpd  * Time.deltaTime;
                 yawRate = Mathf.Clamp(yawRate, -60.0f, 60.0f);
                 transform.rotation = Quaternion.Euler(pitchRate, yawRate, 0);
             }
      
         rigidbody.AddRelativeForce(pitch, yaw, shipSpd);
     }