Hi elsaluebing
Joomla 4 uses smart search module called 'mod_finder' instead of old search module.
You can override this module in template by copying the file: root/plugins/system/t3/base-bs3/html/mod_finder/default.php
paste into template folder: root/templates/purity_iii/html/mod_finder/
At approx line 72, add the search icon like this:
<div class="search">
<i class="fa fa-search"></i>
<form class="mod-finder js-finder-searchform form-search" action="<?php echo JRoute::_($route); ?>" method="get" role="search">
...
I updated for you, kindly check.