The absurd word "bilgisayar" in Turkish, was supposed to mean what the term computer does in American/English.
To literally translate, "bilgisayar" means one of
Only the third alternative, is not exactly wrong, although even that case is also not able to capture mainly what the thing computer is or does. The full American/English term does capture the meaning fairly, as a stored-program computer.
I am afraid, we may find yet another explanation. If the absurd translation as with the case of veritabani, is also the case here, we may (wildly) guess that, the coiner translated the "stored program" as "knowledge" and somehow, similarly, tried to approximate the word "compute" with the first [if not only] Turkish-origin word, around -- as the word hesab is from Arabic (even if centuries old), the newspeak people would not want to refer to that, we may infer. (Are the islem, islemci, etc., those from the root is/ish (work), relatively recent?)
Both "bilgi" and "sayar" being out of context, the word is twice absurd.
Bilgi means knowledge, as opposed to a load of newspeak set of words "invented" to rival the American or French words, veri (data), im (sign), simge (symbol), ileti (message), ... (information). The computer only knows of data, to process that data in a deterministic way, as guided by the stored-program -- and that program is data, too.
The word "sayar" (the general tense of the verb saymak) means "counts" or "shows respect to" or "lists (some list)." The rare computer, if at all, may even own an instruction to "count" or to "show respect to" or to "dump to screen" -- to program with. Instead, we employ an increment/decrement or add/subtract instruction, to count. To dump to screen, a program would probably need several instructions, and the monitor/printer/etc. is a peripheral device. That is, even to dump a visible output, the computer is not enough. Not to even mention that we also compute and store data with that gadget.
Such "local" coinage is only low-quality. A tribal "me-too" urge to avoid a foreign term. No reflection. No merit. If I were to tell that word to grandma, would she infer any further, than what the foreign (non-Turkish) word computer could tell?