In a turbulent stream the speed of air changes rapidly in space and time, as it consists of “vortexes” of multiple scale. Air in these vortexes hits your plane from different directions, so for you it feels like a bumpy ride.
As an example of how a turbulent flow *may* look like, see pictures of [Kelvin-Helmholtz instability](https://en.wikipedia.org/wiki/Kelvin%E2%80%93Helmholtz_instability).
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