header and wp_redirect not working. cannot modify header information warning

There is no way do perform a redirect, if you already printed anything to the browser. (To make a redirect you have to send proper header to the browser. And headers have to be sent before any output.)

So if you want to perform any redirects, you can’t (and shouldn’t) put such code directly in template. It should be placed in functions.php and run on some hook.

Also… It’s not the best idea to send any form with no action set – it’s a security problem. And there is already a mechanism built in WP, that you should use for such requests.

So if you want to do this properly, it may look something like this…

In functions.php file:

function process_my_form() {
    if ( ! empty($_POST['password']) ) {
        $redirect = ($_POST['password']);
        $redirect = str_replace(" ","-",$redirect);
        $redirect = strtolower($redirect);
        wp_redirect( "/bids/$redirect" );
        die;
    }

    wp_redirect( '<URL OF YOUR FORM>' );  // redirect back to the form, so replace it with correct URL
    die;
}
add_action( 'admin_post_process_my_form', 'process_my_form' );
add_action( 'admin_post_nopriv_process_my_form', 'process_my_form' );

And in your template file:

<div class="row secureform__container">
    <form id='securedownloads' class="secureform" method='post' action='<?php echo esc_attr( admin_url('admin-post.php') ); ?>'>
        <fieldset class="secureform__inner">
            <label for="password" class="secureform__label">Password:</label>
            <input id='password' type="text" name="password" maxlength="40" class="secureform__input">
            <input type="submit" value="login" class="secureform__submit">
            <input type="hidden" name="next" value="">
            <input type="hidden" name="action" value="process_my_form">
        </fieldset>
    </form>
</div>