Solved, the site's label; the answer is yes, and the standing here is solved and proved, derived from the claim pages below. The site credits the affirmative answer, with the Turán graph as the one exception, to Bollobás and Thomason [BoTh81], credits Bondy [Bo83b] with the strengthening that for more than $\mathrm{ex}(n;K_r)$ edges the vertex can be taken of maximum degree, and the community database records the problem as solved. The two status-defining sources are B. Bollobás and A. Thomason, "Dense neighbourhoods and Turán's theorem", J. Combin. Theory Ser. B 31 (1981), no. 1, 111--114, and J. A. Bondy, "Large dense neighbourhoods and Turán's theorem", J. Combin. Theory Ser. B 34 (1983), no. 1, 109--111 (both refereed, per their Crossref records). Neither source card holds a file. [BoTh81] (library home Bollobás and Thomason 1981, the publisher's open-archive version) carries the affirmative answer in its theorem (printed p. 111), paged at Bollobás and Thomason 1981, Theorem (p. 111), in the paper's indexing with $T_r(n)$ the $r$-partite Turán graph and $t_r(n)$ its number of edges: "Let $G$ be a graph of order $n$ with $m\ge t_r(n)$ edges. Then either $G=T_r(n)$ or else there is a vertex $x$ such that $G[\Gamma(x)]$, the subgraph spanned by the neighbours of $x$, contains at least $t_{r-1}(d)+1$ edges, where $d=d(x)$. Furthermore $d(x)>n(1-1/r-1/(1+\sqrt r))$." With $t_{r-1}(n)=\mathrm{ex}(n;K_r)$ this is the site's statement in the paper's letters, with the "$+1$" of Erdős's question in the conclusion, the Turán graph the only exception, and an explicit constant, $c_r=1-\frac1{r-1}-\frac1{1+\sqrt{r-1}}$ in the site's indexing ($c_4\approx0.30$). The paper attributes the conjecture to [Er75], its [2], in the form "if $e(G^n)>t_r(n)$" (p. 111). Its proof (pp. 112--114) is followed on the card, which records four observations (three misprints and a final inequality that needs $n\ge r(1+\sqrt r)^2/2$ in the paper's $r$). [Bo83b] (library home Bondy 1983, the publisher's version with its erratum) carries Bondy's own strengthening as its Theorem 2 (p. 110), paged at Bondy 1983, Theorem 2, in the same indexing: "Let $G$ be a simple graph on $n$ vertices and more than $t_r(n)$ edges, where $r\ge2$, and let $v$ be a vertex in $G$ of degree $m=\Delta(G)$. Then the subgraph induced by the neighbours of $v$ has more than $t_{r-1}(m)$ edges"; its proof is twelve lines. In the problem's letters: more than $\mathrm{ex}(n;K_r)$ edges, Erdős's $f_r(n)$, force every vertex of maximum degree $m$ to have a neighborhood with more than $\mathrm{ex}(m;K_{r-1})$ edges, Erdős's "at least $f_{r-1}(m)$"; the maximum degree is at least the average degree, which is more than $2\,\mathrm{ex}(n;K_r)/n\ge(1-\frac1{r-1})n-\frac{r-1}{4n}$, so the degree is linear (this page's own observation; the note states no degree bound for its own theorem). The note also states the theorem of [BoTh81] as its Theorem 1 (p. 109), paged at Bondy 1983, Theorem 1, with the attribution "proved independently by Bollobás and Thomason [1] and Erdős and Sós [4]" and the degree bound $d(v)>(1-\frac1r-\frac1{1+\sqrt r})n$ as its (1); the restatement agrees with the 1981 theorem, its "more than $t_{r-1}(m)$ edges" being that paper's "at least $t_{r-1}(d)+1$ edges", and it names a second proof, an Erdős--Sós preprint, which is not among this page's sources. Bondy's examples (p. 111), graphs with exactly $t_r(n)$ edges that are not Turán graphs and whose maximum-degree vertices do not have the strict conclusion, show that the "$>$" of the site's account of [Bo83b] cannot be weakened for that conclusion; for the site's non-strict conclusion a maximum-degree vertex always works, as the Bondy claim page records. Nothing here is independently reviewed. The site's label SOLVED attaches to an affirmative answer proved in two refereed notes; the claim pages (Bollobás and Thomason, full, and Bondy, partial, for graphs with more than $\mathrm{ex}(n;K_r)$ edges) record the result as proved.