My dad, a consummate gardener, has a long-established bed of horseradish. He yanks it out when it spreads too far. Anyway, I believe the OP said that this is first-year growth. That’s your problem right there.
When it was my job to go out & pick a root to grate into the nightly salad, I noticed right away that the older, thicker roots were much stronger than the new, pencil-thin ones. If you can put up with the native unruliness of horseradish in your garden, it will get stronger over time.