To understand this concept, consider the buoyant force acting on the bucket. Buoyant force is the upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In the case of a bucket of water submerged in water, the buoyant force acting on the bucket is equal to the weight of the water displaced by the bucket.
Since the bucket and the water inside it are both made of water, they have the same density as the surrounding water. This means that the weight of the water displaced by the bucket is exactly equal to the weight of the bucket and the water it contains. Therefore, the bucket does not become lighter under water.