Have you noticed that whenever you fill your bathroom sink a leak develops under the sink onto the cabinet floor? You probably need to be looking at your overflow.
On some bathroom sinks the overflow is integral to the sink, but on most sinks, you will find that the overflow drain is basically not part of the basin, rather it attaches to the underside of the sink and finally to the drain assembly.
In many cases the overflow is just a molded piece of plastic that attaches to the bowl.
When water enters the overflow hole near the rim of the sink, it is guided down this piece of plastic and to the drain assembly. Since it isn’t often (or shouldn’t be that often) that water enters the overflow, you probably wouldn’t have the need to be concerned, but these plastic parts traditionally don’t seal very well to the underside of the sink.
In fact, you may be able to see gaps along the underside of the sink where it connects.
You can try to fill in any gaps with silicone caulking, but don’t expect much. It may work well enough since the water should be infrequent. If you find that the problem continues, you probably will need to replace the sink.