That's pretty fascinating.
 
Instagram is actually currently building a new update for Threads. This update will mean that still image posts that are shared from IG will be shown in a new, dedicated format rather than as a link with a small thumbnail, which tends to push the user over to IG.
The new process will display a single image and carousel posts from Instagram in full size in your Threads updates, overlaying the creators' Instagram handle on the image. Tap on a picture and you'll find yourself through to the original post on Instagram.
 
The same presentation is being tested for Reels clips, presenting Instagram a much more integrated, and visually pleasing share format for its content in its sister app.

Which may be a fantastic way to expand both your Threads and your IG by republishing your updates to Threads, cutting time that you may have had otherwise to create, while still appearing as a reasonable-looking Threads profile display.
 
For now, though it isn't in all users' hands. I wasn't able to get this to occur in my own posting tests, but it seems to be coming for all users sometime soon.

We have reached out to the Threads team in order to get more information about the rollout of the new link of the new presentation of IG.