Pagination replacing new search results with old ones when try to change page

I've added a ajax search box in app's home page. It works fine. But when i use the pagination after the new results shows up, pagination leads me to the old results. I'm not sure what I'm not doing right. So looking for expert's help to get a solution for it.

I've these in routes --

Route::get('/', 'BookmarkController@index');
Route::post('/', 'BookmarkController@search');

I've these in their controller -

public function index(Request $request)
{
    $tags_list = Tag::orderBy('tag', 'asc')->get();
    $bookmarks = Bookmark::orderBy('created_at','desc')->where('public', '1')->paginate(10);
    $bookmarks_all = Bookmark::orderBy('created_at','desc')->where('public', '1')->get();
    return view('welcome')->with('bookmark', $bookmarks)->with('tags_list', $tags_list)->with('bookmarks_all', $bookmarks_all);
}

public function search(Request $request){
    $search_value = $_POST['search'];
    
    $bookmarks = Bookmark::orderBy('created_at','desc')->where('public', '1')->where('title', 'rlike', $search_value)->orwhere('description', 'rlike', $search_value)->orwhere('contents', 'rlike', $search_value)->orwhere('tags', 'rlike', $search_value)->paginate(10);
    return view('public_bookmarks')->with('bookmark', $bookmarks);
}

In welcome.blade.php

<div class="container content-container">
<div class="row ">
    <div class="col-sm-12">
        <div class="page-header">
            <div class="row">
                <span class="title col-sm-8">Recent Bookmarks</span>
                <form id="demo-2" class="search-form col-sm-4" method="post">
                    {{ csrf_field() }}
                    <div class="input-group">
                        <!-- <input id="search" class="form-control" onkeyup="search_data(this.value, 'result');" placeholder="Search" name="search" value="" type="text"> -->
                        <input id="search" class="form-control" placeholder="Search" name="search" value="" type="text">
                        <span class="input-group-btn">
                            <button class="btn btn-primary" id="search-btn" type="button"><i class="fa fa-search"></i></button>
                        </span>
                    </div>
                    <script type="text/javascript">
                        $(document).ready(function() {
                            $('#search-btn').click(function(){
                                $.ajax({
                                    url: '/',
                                    type: "post",
                                    data: {'search':$('input[name=search]').val(), '_token': $('input[name=_token]').val()},
                                    success: function(data){
                                        $('#search-results').html(data);
                                    },

                                    error: function (data) {
                                        console.log('Error on Article extracting');
                                        console.log(data);
                                    }
                                }); 
                            });
                        });
                    </script>
                </form>
            </div>
        </div>
    </div>
    <div class="col-sm-9" id="search-results">
        
        @include ('public_bookmarks')
    </div>
</div>

In public_bookmars.blade.php

@if (count($bookmark) > 0)
<div class="row card-row">
@foreach ($bookmark as $bookmark_single)
    <div class="col-sm-4 col-xs-12 card-parent" data-col="col-sm-4">
        <div class="card">
            <div class="card-part1">
                <div class="img-card">
                    <img src="{{$bookmark_single->thumbnail}}" />
                </div>
                <div class="card-content">
                    <h4 class="card-title">
                        {{ $bookmark_single->title }}
                    </h4>
                    <div class="card-desc">
                        {{ str_limit($bookmark_single->description, $limit = 50, $end = ' [...]') }}
                    </div>
                </div>
                <div class="card-read-more">
                    <p><?php   $tags = $bookmark_single->tags;
                        $tag_list = explode(',', $tags); ?>
                        @foreach ($tag_list as $tag)
                            <a href="/tag/{{$tag}}" class="label label-primary">{{$tag}}</a>
                        @endforeach
                    </p>
                    <p class="card-user">- &nbsp;<a href="/user/{{ $bookmark_single->bookmarker }}">{{ $bookmark_single->bookmarker }}</a></p>
                    <a class="v-link" target="_blank" href="{{ $bookmark_single->url }}">Visit the link</a>
                </div>
                <button type="button" class="btn btn-success btn-circle btn-lg btn-read-more"><i class="fa fa-chevron-right"></i></button>
            </div>

            <div class="card-part2 col-xs-0">
                {{ print $bookmark_single->contents }}
            </div>
        </div>
    </div>
@endforeach
</div>
{{ $bookmark->links() }}
@endif