The RNA World hypothesis posits that RNA was the molecule of both heredity and function during the emergence of life. This hypothesis implies that RNA templates can be copied, and ultimately replicated, without the catalytic aid of evolved enzymes. A major problem with nonenzymatic template-directed polymerization has been the very poor copying of sequences containing rA and rU. Here, we overcome that problem by using a prebiotically plausible mixture of RNA mononucleotides and random-sequence oligonucleotides, all activated by methyl isocyanide chemistry, that direct the uniform copying of arbitrary-sequence templates, including those harboring rA and rU. We further show that the use of this mixture in copying reactions suppresses copying errors while also generating a more uniform distribution of mismatches than observed for simpler systems. We find that oligonucleotide competition for template binding sites, oligonucleotide ligation and the template binding properties of reactant intermediates work together to reduce product sequence bias and errors. Finally, we show that iterative cycling of templated polymerization and activation chemistry improves the yields of random-sequence products. These results for random-sequence template copying are a significant advance in the pursuit of nonenzymatic RNA replication.