Motivated by problems in topology, we explore the complexity of balanced group presentations. We obtain large lower bounds on the complexity of Andrews-Curtis trivialisations, beginning in rank 4. Our results are based on a new understanding of how Dehn functions of groups behave under certain kinds of push-outs. We consider groups $S$ with presentations of deficiency 1 satisfying certain technical conditions and construct balanced group presentations $\P_w$ indexed by words $w$ in the generators of $S$. If $w=1$ in $S$ then $\P_w$ is Andrews-Curtis trivialisable and the number of Andrews-Curtis moves required to trivialise it can be bounded above and below in terms of how hard it is to prove that $w=1$ in $S$.