What can be done to correct my bulging eyes? Are they normal? (Photos)

You likely have inherited bulging eyes as part of normal variant but see an oculoplastic specialist for evaluation. There are various treatment options avaiable based on finding in consultation.

A combination approach will give the most improvement. Fillers, lasers, and microneedling/PRP can improve under the eyes. My patients try to get non surgical treatments to avoid surgery. We use PRP and stem cells there often. I suggest consulting with an expert to develop a treatment plan best suited for your short and long term goals. Best, Dr. Emer

I don't think your eyes are bulging based on your photos.  I do think you have a some undereye volume loss which could be treated using a filler like Restylane. This would help with the undereye dark circles and shadowing.

Hello, and sorry to hear you have suffered from this so long.  I believe you have beautiful eyes and if you want fillers under eyes you can try them. Your eyes look normal, hope that TSH was normal when it was checked, otherwise know that you have perfectly normal eyes and may have little undereye fat which is cosmetic minor procedure  with fillers. Hope that helps, Dr Violetta Mailyan.

Thank you for your question.  Looking at your picture I believe that adding volume to the cheeks would help balance your look.  By creating fullness to the midface the eyes can appear less prominent.  I also think you could benefit from tear trough filler to again soften the transition from the eyes to the cheek.  
Nice options for the tear trough area are PRP or restylane.  For the cheeks you can really use most fillers, it just depends on how long you'd like the product to last.  Some of the more popular midface fillers are Voluma, Lyft, Bellafill, Sculptra and microfat transfer.