  In 1964, Golod and Shafarevich found that, provided that the number of
relations of each degree satisfy some bounds, there exist infinitely
dimensional algebras satisfying the relations. These algebras are called
Golod-Shafarevich algebras. This paper provides bounds for the growth function
on images of Golod-Shafarevich algebras based upon the number of defining
relations. This extends results from [32], [33]. Lower bounds of growth for
constructed algebras are also obtained, permitting the construction of algebras
with various growth functions of various entropies. In particular, the paper
answers a question by Drensky [7] by constructing algebras with subexponential
growth satisfying given relations, under mild assumption on the number of
generating relations of each degree. Examples of nil algebras with neither
polynomial nor exponential growth over uncountable fields are also constructed,
answering a question by Zelmanov [40].
  Recently, several open questions concerning the commutativity of algebras
satisfying a prescribed number of defining relations have arisen from the study
of noncommutative singularities. Additionally, this paper solves one such
question, posed by Donovan and Wemyss in [8].
