We have studied four complex organic molecules (COMs), the oxygen-bearing methyl formate (CH_3_OCHO) and dimethyl ether (CH_3_OCH_3_) as well as the nitrogen-bearing formamide (NH_2_CHO) and ethyl cyanide (C_2_H_5_CN), towards a large sample of 39 high-mass star-forming regions representing different evolutionary stages, from early to evolved phases. We aim to identify potential correlations and chemical links between the molecules and to trace their evolutionary sequence through the star formation process. We analysed spectra obtained at 3, 2, and 0.9mm with the IRAM-30m telescope. We derived the main physical parameters for each species by fitting the molecular lines. We compared them and evaluated their evolution while also taking several other interstellar environments into account. We report detections in 20 sources, revealing a clear dust absorption effect on column densities. Derived abundances range between ~10^-10^-10^-7^ for CH_3_OCHO and CH_3_OCH_3_, ~10^-12^-10^-10^ for NH_2_CHO, and ~10^-11^-10^-9^ for C_2_H_5_CN. The abundances of CH3OCHO, CH3OCH3, and C2H5CN are very strongly correlated (r>=0.92) across 4 orders of magnitude.We note that CH_3_OCHO and CH_3_OCH_3_ show the strongest correlations in most parameters, and a nearly constant ratio (1) over a remarkable 9 orders of magnitude in luminosity for the following wide variety of sources: pre-stellar to evolved cores, low- to high-mass objects, shocks, Galactic clouds, and comets. This indicates that COMs chemistry is likely early developed and then preserved through evolved phases. Moreover, the molecular abundances clearly increase with evolution, covering 6 orders of magnitude in the luminosity/mass ratio. We consider CH_3_OCHO and CH_3_OCH_3_ to be most likely chemically linked. They could, for example, share a common precursor, or be formed one from the other. Based on correlations, ratios, and the evolutionary trend, we propose a general scenario for all COMs, involving a formation in the cold, earliest phases of star formation and a following increasing desorption with the progressive thermal and shock-induced heating of the evolving core.