The way your links are displayed on Twitter depends on whether or not Twitter Cards are enabled on the website. Twitter Cards are an alternative format for tweets, where rich photos, videos and media experiences are attached to your tweet in a more attention grabbing fashion, helping to drive traffic to the website. Learn more about Twitter Cards here.
With that being said, tweets that are link post, will be posted with Twitter Cards if Twitter Cards are enabled on the website, as shown below.
Now, if Twitter Cards are not enabled on the website, the tweet will be posted as a regular tweet, without an image.
Keep in mind that Twitter Cards can’t be added within Postcron or directly on Twitter by the end user, as they are automatically generated for your tweets if Twitter Cards have already been enabled on the website that you're sharing a link to.
If you need to preview how a link you're planning to share will be displayed once posted, you can use Twitter’s Card Validator tool.