<style>
  * {
    list-style-type: none;
    margin: 0;
    padding: 0;
  }

  div, li {
    border: 1px solid rgb(142, 28, 47);
  }

  .header {
    display: grid;
    grid-template-columns: 5fr 1fr 1fr;
  }

  ul {
    display: flex;
    justify-content: space-between;
    margin-top: 10px;
  }
</style>

<div class="header">
  <div class="Search">Search</div>
  <div class="Alert">Alert</div>
  <div class="User Info">User Info</div>
  <div class="Intro">Hi There</div>
  <ul>
    <li>New</li>
    <li>Upload</li>
    <li>Share</li>
  </ul>
</div>
justify-items controls how items are aligned inside each individual grid cell, not how the items themselves are spaced in the container.
space-between is not a valid value for justify-items.