  Given a graph $H$, a balanced subdivision of $H$ is a graph obtained from $H$
by subdividing every edge the same number of times. In 1984, Thomassen
conjectured that for each integer $k\ge 1$, high average degree is sufficient
to guarantee a balanced subdivision of $K_k$. Recently, Liu and Montgomery
resolved this conjecture. We give an optimal estimate up to an absolute
constant factor by showing that there exists $c>0$ such that for sufficiently
large $d$, every graph with average degree at least $d$ contains a balanced
subdivision of a clique with at least $cd^{1/2}$ vertices. It also confirms a
conjecture from Verstra{\"e}te: every graph of average degree $cd^2$, for some
absolute constant $c>0$, contains a pair of disjoint isomorphic subdivisions of
the complete graph $K_d$. We also prove that there exists some absolute $c>0$
such that for sufficiently large $d$, every $C_4$-free graph with average
degree at least $d$ contains a balanced subdivision of the complete graph
$K_{cd}$, which extends a result of Balogh, Liu and Sharifzadeh.
