Sure, but why is it pulled sideways and not down, and why does it move toward the river of all directions it could move?

For exactly the same reason the river exists in the first place. Why does the water flow to the ocean and not just sink into the ground?

A lot of it does sink into the ground though. Thats how aquifiers replenish.

I think the percentage of fertilizer in the groundwater would reach an equilibrium where just as much is going out as is coming in.

Water from shallow aquifers also flows into nearby rivers.

Some water is pulled down (along with the half of the fertilizer that's actually used), the rest is literally washed away because the soil can't absorb the amount of water being dropped on it. All the water that's flowing over the ground ends up in rivers (it's the reason they exist in the first place).

Some of it does I am sure, where some will be consumed by soil bacteria. Lots of fertilizers are salts as well and even if it goes into the soil, rain will redissolve it where it will eventually make its way to rivers.