When your bathroom sink smells like rotten eggs when you run water it can be a result of sulfur in the water or sulfur from the water heater.
A rotten egg smell from your bathroom sink can also be from mold growth in the sink or sewer gas coming up from another drain.
A blocked drain vent can also cause your bathroom sink and drains to smell like rotten eggs or sewer.
other causes of a rotten egg smell in your sink drain are evaporated water in the P trap, clogged drains or bacteria.
If the water in the sinks P trap has evaporated then it can remove the barrier which prevents the sewer gas from coming up.
Bacteria can grow in the sink drain from organic matter like soap, hair and food and can produce a gas which smells like sewage or rotten eggs.
If you have well water then the well water can also contain sulfur which can lead to a rotten egg smell in the sink.
But the water will also smell like sulfur in other sinks.
If it's just the bathroom sink then it's likely from the bathroom sink drain causing the rotten egg smell.
You can help get rid of the rotten egg smell in your sink drain by pouring some baking soda and vinegar down the drain and allow it to sit.
Then after 30 minutes to an hour or so you can run some hot water down the drain.