Chaperones are essential to the co-translational folding of most proteins. However, the principles of co-translational chaperone interaction throughout the proteome are poorly understood, as current methods are restricted to few substrates and cannot capture nascent protein folding or chaperone binding sites, precluding a comprehensive understanding of productive and erroneous protein biosynthesis. Here, by integrating genome-wide selective ribosome profiling, single-molecule tools, and computational predictions using AlphaFold we show that the binding of the main E. coli chaperones involved in co-translational folding, Trigger Factor (TF) and DnaK correlates with "unsatisfied residues" exposed on nascent partial folds - residues that have begun to form tertiary structure but cannot yet form all native contacts due to ongoing translation. This general principle allows us to predict their co-translational binding across the proteome based on sequence only, which we verify experimentally. The results show that TF and DnaK stably bind partially folded rather than unfolded conformers. They also indicate a synergistic action of TF guiding intra-domain folding and DnaK preventing premature inter-domain contacts, and reveal robustness in the larger chaperone network (TF, DnaK, GroEL). Given the complexity of translation, folding, and chaperone functions, our predictions based on general chaperone binding rules indicate an unexpected underlying simplicity.