What word describes the melancholy felt at a place of great loss?

I'll throw 'haunted' into the mix. Not haunted in the sense of "visited by spirits, imaginary beings, apparitions, spectres, etc.", although that common meaning may figure in the background of the sense conveyed, but rather haunted in this transfigurative sense:

Of memories, cares, feelings, thoughts: To be visited by these; to have these come up or present themselves as recurrent influences or impressions, esp. as causes of distraction or trouble.

(Paraphrased from the OED, haunt, v.)

This word works very well in your example:

I felt haunted when visting Normandy. Thinking of how many people died there hit me hard.

Somber communicates a sense of seriousness and depression:

a : of a serious mien : grave < somber dignitaries >
b : of a dismal or depressing character : melancholy (MW)

Mournful emphasizes the sorrow:

Feeling or expressing sorrow or grief; sorrowful. (TFD)

the mournful survivors of the disaster were faced with the grim task of burying the dead (MW)

How about despondent?

feeling or showing extreme discouragement, dejection, or depression (MW)

I think this word also conveys the despair one feels when realizing that so much tragedy is inflicted by fellow human beings. Many places of great loss are testaments to that terrible truth.