Alluvium is anything carried by a river. It will only be carried in times of flood, and it will settle to the bottom when the water stops flowing quickly. This is why rivers are a brown colour in flood, but clear at other times.

When a flooded river enters a lake, it will be full of alluvium, but the lake waters are still. As a result, the alluvium will settle quickly close to the river mouth, and this is why it builds up by the lake edge.

Eventually the alluvium builds up so much it reaches the surface, and that is when you see the fan pushing out into the lake.