The site's wording admits every bipartite graph, and its first question fails at the smallest ones. A graph $G$ with one edge, $K_2$ together with any isolated vertices, has $\mathrm{ex}(n;G)=0$ for $n\ge|V(G)|$, since copies need not be induced and a host avoiding $G$ then has no edges; so for every $\alpha\in[1,2)$ and $c>0$ the ratio $\mathrm{ex}(n;G)/(cn^\alpha)$ is eventually $0$ and does not tend to $1$. A graph with no edges is contained in every host with at least $|V(G)|$ vertices, so its extremal number is undefined for large $n$. These checks are the corpus's own. The failures lie at the smallest sizes of the forbidden graph, the graphs with at most one edge, and at these sizes no graph can meet the conclusion. The next size has none: a bipartite graph with two edges is a path with two edges or two disjoint edges, with isolated vertices added, and for large $n$ its extremal number is $\lfloor n/2\rfloor$ (a perfect or near-perfect matching) or $n-1$ (a star), so $\mathrm{ex}(n;G)\sim cn$ with $\alpha=1$.

The change inserts "with at least two edges" after "every bipartite graph $G$"; nothing else changes. It is the corpus's own correction, excluding exactly the forbidden graphs too small for the conclusion to hold. Erdős's own statement of the conjecture with Simonovits ([Er67d], p. 119, display (8); library card) does not fail at the graphs with one edge: it is written for the least number $f(n;G)$ of edges that forces $G$, which is $\mathrm{ex}(n;G)+1$, with $\lim f(n;G)/n^\alpha=c>0$ for some $0\le\alpha<2$, and a graph with one edge has $f(n;G)=1$ for large $n$. The failure there comes from the site's restatement in terms of $\mathrm{ex}$ with $\alpha\in[1,2)$. The formal-conjectures statement file adopts the same two-edge condition.