Can Veneers Hide Thininng Enamel?

Many dentists that discourage veneers don't know how to do them and don't understand their application. Depending on how much tooth structure remains, the bite forces and the condition of the remaining tooth structure (some teeth don't bond very well) veneers may work very well.
If not, full coverage crowns may be indicated.
Whichever is chosen, be sure all porcelain materials are used, not restorations with metal underneath.

If the front of your tooth is thin and there is enough enamel then you can get veneers. 

If the back of your tooth is damaged and thin then I would recommend a crown to cover your entire tooth.

If the thinning is the result of loss of enamel on the inside, then veneers will not solve the problem. You are better off with a full coverage option of crowns.

You mentioned your two front teeth are extremely thin. Your dentist does not recommend veneers because the problem is in the back of the tooth.  Veneers are a wonderful choice to give you two very beautiful front teeth.  They wrap around the tooth but do not totally cover the back.  No one can see the back, so this should not be the cause of your embarrassment. Veneers would be less bulky than crowns or caps and your smile would be more natural looking and cosmetically pleasing.

Veneers are designed to replace the enamel on teeth that is missing or damaged. Even though the veneer does not cover the enamel very much on the back of the tooth, it will to some extent. I believe them to be a better option than crowns or caps that can make the teeth look big and bulky if done improperly. If you do nothing and end up fracturing your teeth off, then root canal treatment and crowns will be your only choice. Do veneers first to strengthen the teeth back and then get a night guard to protect the remaining enamel.
Dr Thomas