Hide URL in final tweet and leave only the card - php

Question

I've developed an app in Laravel, using codebird-php, that's able to post tweets with images, videos and links. The code works as it's intended but I was told by my boss to hide the url in the final tweet and leave only the card, this for aesthetic purposes.
You can see what I'd like to do in the next pictures.
On top you can see a tweet from TechCrunch with a card but no url in body, this is wat I want to achieve.
However, I've also noted that there are tweets with cards and a visible url in the body. TechCrunch also as an example.
I've read into Twitter's API thinking the url might be an additional parameter as Facebook does, but I found nothing. I've also thought that it might be Twitter's site hiding the urls in web as in Twitter for Mac I can see the url.
I really don't know, I've been looking for a couple of days now and don't know what or how to do it. Does anyone know how to achieve that? Or am I right thinking that it's Twitter hiding the url in web?
Thank you all for your time.
EDIT:
Here's my code.
$sm = SocialScheduling::find($id);
$sm->update([
'persona_id' => $request->get('persona_id'),
'social_auth_id' => $request->get('social_auth_id'),
'social_scheduling_category_id' => $request->get('social_scheduling_category_id'),
'message' => $request->get('message'),
'post_link' => $request->get('post_link'),
'post_to_feed' => $request->get('ptf'),
'social_networks' => json_encode($request->get('social_networks')),
'facebook_pages' => json_encode($request->get('facebook_pages')),
'programmable_type' => $request->get('programmable_type'),
'programmable_id' => $request->get('programmable_id'),
'client_id' => $request->get('client_id'),
'image_url' => $request->get('image_url'),
'publish_time' => $request->get('publish_time')
]);
$sm->save();
$sm->load('persona','client','category','programmable','social_auth');
$persona = SocialAuth::find($sm->social_auth_id);
Codebird::setConsumerKey( Config::get('services.twitter.client_id'),
Config::get('services.twitter.client_secret'));
$tw = Codebird::getInstance();
$tw->setToken($persona->twitter_token,$persona->twitter_token_secret);
$data = [
'status' => $sm->message
];
$reply = $tw->statuses_update($data);
What my code does is it updates the SocialScheduling model, then it uses social_auth_id to find the model with the credentials and assigns it to $persona, afterwards I build the tweet adding the url to the tweet message, but I'd like to know if there's an additional parameter, like Facebook does, where I can send the url. I didn't see anything in Twitter's docs.