Hydrogels are used in a wide range of biomedical applications, including three-dimensional (3D) cell culture, cell therapy and bioprinting. To enable processing using advanced additive fabrication techniques and to mimic the dynamic nature of the extracellular matrix (ECM), the properties of the hydrogels must be possible to tailor and change over time with high precision. The design of hydrogels that are both structurally and functionally dynamic, while providing necessary mechanical support is challenging using conventional synthesis techniques. Here, we show a modular and 3D printable hydrogel system that combines a robust but tunable covalent bioorthogonal cross-linking strategy with specific peptide-folding mediated interactions for dynamic modulation of cross-linking and functionalization. The hyaluronan-based hydrogels were covalently cross-linked by strain-promoted alkyne-azide cycloaddition using multi-arm poly(ethylene glycol). In addition, a de novo designed helix-loop-helix peptide was conjugated to the hyaluronan backbone to enable specific peptide-folding modulation of cross-linking density and kinetics, and hydrogel functionality. An array of complementary peptides with different functionalities was developed and used as a toolbox for supramolecular tuning of cell-hydrogel interactions and for controlling enzyme-mediated biomineralization processes. The modular peptide system enabled dynamic modifications of the properties of 3D printed structures, demonstrating a novel route for design of more sophisticated bioinks for four-dimensional bioprinting.