Once upon a time, when you posted a photo on Instagram, you could also post it as a native photo on Twitter (i.e. as a photo post and not as a link).
When Facebook acquired Instagram, things got a bit unfriendly and Twitter no longer allowed native posts from Instagram.
Therefore, if you try to post to Twitter using the Instagram app, what you will see is a URL post.
Use IFTTT to post from Instagram to Twitter – that way you will get nice photo posts, not link posts.
Check that you don’t have circular references with IFTTT. I once got caught with a IFTTT to Facebook anything I posted on Tumblr but the Tumblr setting to also post any link on Facebook. Within hours I had hundreds of reposts / link. 😦 (Same with Pinterest – check that your Facebook / Twitter settings are turned off if you’re using IFTTT)