My eyelids always feel heavy. Any suggestions to fix them? (Photo)

This image and your story is simply not sufficient to conclude that you have exophthalmos let alone thyroid related exophthalmos. I would encourage you to be assessed by a caring oculoplastic surgeon who is also board certified ophthalmologist to determine the basis for your concerns.

You may have an eye configuration called exophthalmos, where the eye protrudes. Sometimes there is a medical cause for this such as too much thyroid hormone. An internal medicine doctor would be a good first start to have a physical and lab tests to make sure your levels OK. The configuration of a forward set globe makes the upper eyelid tighter on the globe, creating a sensation of heaviness or difficulty closing. In patients with severe exophthalmos, there may be issues with the cornea becoming dry, causing visual issues.

It is very difficult to tell with the limited view you have submitted. But it appears that your problem may be "Exophthalmos".. That is a condition frequently associate with Hypothyroidism. You may want to consult with an Ophthalmologist (eye doctor) first , to assess what the problem is..