Your mail client is likely blocking remote content by default. On macOS Mail you should see a banner stating as such at the top.
If that isn't the problem, and certain images are still not showing, then the likelihood is that these images are actually attachments.
If you try to access that attachment as a guest (or in a browser where you're logged out), that will give you an idea as to what is going on. Your browser needs to be logged in to be able to view attachments.
Your mail client isn't logged into the forum, so it can't display the image. On the other hand, your webmail is likely in the same browser where you are logged in so the images will display correctly there.