How can I get natural gum tissue over a dental bridge that was placed due to missing tooth and bone loss? (Photo)

you will need procefures  called soft tissue / hard tissue ridge augmentation , and then pontic site preperation with these procedures the area if concern will be " plumped up " with Tissue and bone. Then the shape of the new crown will be carved out in the gum. So the new crown will have an illusion of growing out of the gum. for best results the bridge will need to be removed. The area of concern treated. Then a new bridge will need to    made with a tooth that is shorter than the one that is bothering you in the existing bridge. 

It would be difficult to get gum to grow over the synthetic material that comprises the bridge. You would have to consider adding new or natural gum tissue in conjunction with a new bridge that would have new contours to accommodate the augmentation resulting from any gum reconstruction procedures.

This a good question and your photo
is helpful. As a periodontist, or gum specialist, I use gum tissue grafts to
replace gum tissue that for one reason or another has receded from teeth. Bone
loss and the position of your tooth appear to be causing the condition that
concerns you. Gum grafting can augment the tissue and create the symmetry and
"natural fit" that you mention. The first step is to find a
qualified, experienced periodontist who can perform an exam and recommend a
specific treatment plan so you can again be confident in your smile. Good luck!