1. We are given two functions: $f(x) = \log_3(2x+1)$ and $g(f(x)) = 4x^2$. We need to find the function $y = g(x)$.\n\n2. The problem states that $g$ is composed with $f$, so $g(f(x)) = 4x^2$. This means if we let $u = f(x)$, then $g(u) = 4x^2$.\n\n3. Since $u = f(x) = \log_3(2x+1)$, we can rewrite this as $u = \log_3(2x+1)$. To express $x$ in terms of $u$, we use the definition of logarithm: $$2x+1 = 3^u.$$\n\n4. Solve for $x$: $$2x = 3^u - 1 \implies x = \frac{3^u - 1}{2}.$$\n\n5. Substitute $x$ back into $g(u) = 4x^2$: $$g(u) = 4 \left( \frac{3^u - 1}{2} \right)^2 = 4 \cdot \frac{(3^u - 1)^2}{4} = (3^u - 1)^2.$$\n\n6. Replace $u$ by $x$ to write $g(x)$ explicitly: $$g(x) = (3^x - 1)^2.$$\n\nFinal answer: $$\boxed{g(x) = (3^x - 1)^2}.$$