On my self-hosted WordPress site running 4.9.6 and Bridgy 1.4.1, I have "Bridgy Publish to Twitter" enabled. The excerpt is left blank. Bridgy publishes the Tweet with the title text of the post along with the featured image for the post. However, the link to the post is not posted along with the tweet.
Post example here: http://184.108.40.206/web-lost/
My manual tweet: https://twitter.com/khurtwilliams/status/997429855775084546
Bridgy Twitter user page: https://brid.gy/twitter/khurtwilliams
Bridgy log: http://bit.ly/2IPXef5
Another example is below. I posted an original u-in-reply-to via my website with the same content as this issue entry included here. See the bottom where it says (Originally published at: Twitter)