  A long-standing conjecture of Erd\H{o}s and Simonovits asserts that for every
rational number $r\in (1,2)$ there exists a bipartite graph $H$ such that
$\ex(n,H)=\Theta(n^r)$. 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 Erd\H{o}s 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 Erd\H{o}s 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 Tur\'an's number of $\theta$-graphs
in an asymmetric setting and employ this result to obtain yet another new
rational exponent for Tur\'an exponents; $r=7/5$.
