Just stick your hand out a car window at sixty miles an hour and twist it up and down. Almost any shape will create some lift. The exact cross section simply determines how efficiently it will do so and how it behaves at different angles of airflow. The next thing you need to forget is Bernoulli. At least for a while.
Airplanes stay up because the wings move air downwards.
After a plane goes by, there is a lot of air closer to the ground than there was before. The reaction to moving the weight of this air counteracts the pull of gravity. Blow up a balloon, point the neck down, and let it go. The air that was inside will now be closer to the floor. The balloon goes the other way.