The results of osmosis are different in plant and animal cells.
Plant cells have a strong cellulose cell wall on the outside of the cell membrane. This supports the cell and stops it bursting when it gains water by osmosis.
Water enters the cell by osmosis. The cytoplasm pushes against the cell wall and the cell becomes turgid.
Water leaves the cell by osmosis. The cytoplasm pulls away from the cell wall (plasmolysis) and the cell becomes flaccid and the plant wilts.
Turgid plant cells play an important part in supporting the plant.
Animal cells do not have a cell wall. They change size and shape when put into solutions that are at a different concentration to the cell contents.
For example, red blood cells:
These things do not happen inside the body. Osmoregulation involving the kidneys ensures that the concentration of the blood stays about the same as the concentration of the cell contents.