You can just use the window.location.href when you submit your signup form. So your code for registerUser(...) would look something like this.

async function registerUser(event){
      event.preventDefault()
      const username = document.getElementById('user').value
      const password = document.getElementById('password').value
      const password_confirmation = document.getElementById('password_confirmation').value
      const phone = document.getElementById('tel').value

      const result = await fetch('/register',{
         method: 'POST',
         headers: {
           'Content-Type': 'application/json'
         },
         body: JSON.stringify({
           username,
           password,
           password_confirmation,
           phone
         })
      }).then((res) => res.json())

      if (result.status === 'ok') {
         alert('success');
         // on a successful signup, redirect user to the login page
         window.location.href = 'LINK_TO_SIGNIN_PAGE'; // exp. http://localhost/signin.html
      }else {
        alert (result.error)
      }
    }

Give it a go!

You need to add a redirect here to this line here

for example :

window.location.href = 'login.html'