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Let f^{(r)}(n;s,k) be the maximum number of edges of an r-uniform hypergraph on n vertices not containing a subgraph with k edges and at most s vertices. In 1973, Brown, Erdős, and Sós conjectured that the limit \begin{equation*} \lim _{n\to \infty } n^{-2} f^{(3)}(n;k+2,k) \end{equation*} exists for all k and confirmed it for k=2. Recently, Glock showed this for k=3. We settle the next open case, k=4, by showing that f^{(3)}(n;6,4)=\left (\frac {7}{36}+o(1)\right)n^2 as n\to \infty. More generally, for all k\in \{3,4\}, r\ge 3 and t\in [2,r-1], we compute the value of the limit \lim _{n\to \infty } n^{-t}f^{(r)}(n;k(r-t)+t,k), which settles a problem of Shangguan and Tamo. [ABSTRACT FROM AUTHOR]