I think that avert and divert come from the same source and seem to suggest the change of course whereas avoid is like reducing the chance of the event from happening or abstain or prevent something from happening.

Just to add to the above, avert is almost always used when speaking about your eyes and looking away. Its very rare to hear it used in other situations. Avert is physically doing something like, moving your eyes to look away or turning your car sharply so you don't drive into another car. Avoid is the result of this physical action; so you would avert your eyes to avoid looking at something. Using avert except from talking about your eyes is rare, instead we would say I swerved to avoid hitting the car I took a different route to avoid the traffic. I hope that is clear, English can be hard to explain sometimes. There is so many different ways to say the same thing it is confusing sometimes. Paul

To elaborate on Tim's answer: avert means to turn away, like avert your eyes Avoid means that you completely stay away from someone/something. If you avoided someone on the street, you would either walk down another street or cross the street and stay far away from them.

Imagine that there is someone approaching on the street and you don't wish to speak to this person. You might avert your eyes in order to avoid speaking with her. This means you would turn your eyes to the side ( avert ) in order to prevent interacting with her ( avoid ). I hope this helps!