Use session to change Nav and implement logout.

src/routes/__layout.svelte

@@ -1,14 +1,23 @@
+<script lang="ts">
+  import { session } from '$app/stores'
+</script>
+
 <svelte:head>
   <title>SvelteKit Auth</title>
 </svelte:head>
 
 <nav>
   <a href="/">Home</a>
+
+  {#if !$session.user}
     <a href="/auth/login">Login</a>
     <a href="/auth/register">Register</a>
+  {/if}
 
+  {#if $session.user}
     <a href="/protected">Admin</a>
     <a href="/auth/logout">Log out</a>
+  {/if}
 </nav>
 
 <slot />

src/routes/auth/logout/index.ts

@@ -0,0 +1,15 @@
+import type { RequestHandler } from '@sveltejs/kit'
+import * as cookie from 'cookie'
+
+export const get: RequestHandler = async () => {
+  return {
+    status: 303,
+    headers: {
+      'Set-Cookie': cookie.serialize('session', '', {
+        path: '/',
+        expires: new Date(0),
+      }),
+      location: '/',
+    },
+  }
+}