When you want to indicate that someone is trying hard to avoid looking at something, there are a number of synonyms for the phrase "keep one's eyes off." One possibility is to use the phrase "avert one's gaze," which suggests deliberate efforts to avoid eye contact. Another option is the verb "avoid," as in "he did his best to avoid looking at the gruesome scene." A similar choice is "shun," which implies a deliberate decision to stay away from something, such as "she shunned the horror movie because she couldn't stand the violence." Other synonyms include "ignore," "refrain from," and "abstain from viewing".