The Hebrew word for both exhibit and exhibition is ta-a-ru-KHA.
Taarukha is one of the first neologisms concocted by Eliezer Ben-Yehuda as he revived Hebrew as a spoken language.
The word comes from a-r-kh (ayin-resh-kaf), a very common root that appears in a great multitude of ancient words.
At its most basic, a-r-kh conveys the idea of arranging things in order, albeit sometimes ironically – one uses the verb arakh for setting a table and for waging war.
In the Middle Ages, the same verb began to be applied to editing text as well. In fact, an editor is an orekh, also from the same root.
Hebrew also "estimates" using the same root - you’d say that one ma’arikh that there are so and so jellybeans in a jar, and your estimate would be your ha’arakha.
So how did this root begin to be used for exhibitions?
Eliezer Ben-Yehuda was one of Hebrew's revivers. He invented the word "ta'arukha" for exhibit in 1892.
Back then, an exhibit was called sha'ar ha’matzevah, literally meaning "tombstone gate." Why they called exhibits "tombstone gates," we do not know today. Some people started using the word ma'aracha instead, but that was a sort of generic catchall term sort of meaning array.
In any case Ben Yehuda thought that more appropriate, it seems, and adapted it using basic rules of Hebrew grammar into the word ta'arukha.
In May 1893, when the Great Chicago Fair opened, all the Hebrew newspapers used Ben-Yehuda’s neologism to describe it. Thus the word was disseminated.
In the 20th century, when the students at Jerusalem’s art school Bezalel started having art shows, they used this word for them too. Thus the word came to apply to exhibit as well.
The one thing it doesn't mean is to make an exhibition of oneself. There are any number of words for that in Hebrew, must of which boil down to the Yiddish neologism "schmuck."