Escape stresses the fact of getting away or being passed. If you avoid a person or thing, you keep away from them. To stay away from someone or something:
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To make void or of no effect;. Escape, avoid, evade, elude, shun, eschew mean to get away or keep away from something. To prevent something from happening or to not allow.
When talking to someone, if you avoid the subject, you keep the conversation away from a.