A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph $H$ such that $\ex(n,H)=Θ(n^r)$. So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$, for integers $k\geq 2$. In this paper we add a new form of rationals for which the conjecture is true; $2-2/(2k+1)$, for $k\geq 2$. This in its turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits's conjecture where one replaces a single graph by a family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits's conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon's conjecture. We also prove the an upper bound on the Turán's number of $θ$-graphs in an asymmetric setting and employ this result to obtain yet another new rational exponent for Turán exponents; $r=7/5$.