We investigate structural and thermodynamic properties of surfacegrafted layers of model "bottle-brush" polymers by Monte Carlo simulation. The polymers consist of a longer main chain densely grafted with shorter side chains, of which the latter have some degree of affinity to the surface. Our focus is on the effect of the side-chain surface affinity on the brush properties, which we study in terms of compression isotherms spanning a broad range of grafting densities. For low grafting densities, side-chain adsorption causes the polymers to spread on the surface. As the grafting density is increased, the layer goes through a "pancake-to-brush" transition to form a brush with the main chains aligned perpendicular to the surface. We find that side-chain adsorption is decisive for the structure of dilute layers and in the transition region but has little influence on the properties of dense brushes. The close relation between compression and adsorption isotherms is discussed, and the implications of side-chain adsorption for the ability of the polymer to form a dense brush are investigated. This analysis suggests that side-chain surface affinity alone will not give rise to "brush of bottle-brushes" layers by adsorption of polymers from solution, in agreement with recent experimental results.