The structure of $^{19,20,22}$C has been investigated using high-energy (about 240 MeV/nucleon) one- and two-neutron removal reactions on a carbon target. Measurements were made of the inclusive cross sections and momentum distributions for the charged residues. Narrow momentum distributions were observed for one-neutron removal from $^{19}$C and $^{20}$C and two-neutron removal from $^{22}$C. Two-neutron removal from $^{20}$C resulted in a relatively broad momentum distribution. The results are compared with eikonal-model calculations combined with shell-model structure information. The neutron-removal cross sections and associated momentum distributions are calculated for transitions to both the particle-bound and particle-unbound final states. The calculations take into account the population of the mass $A-1$ reaction residues, $^{A-1}$C, and, following one-neutron emission after one-neutron removal, the mass $A-2$ two-neutron removal residues, $^{A-2}$C. The smaller contributions of direct two-neutron removal, that populate the $^{A-2}$C residues in a single step, are also computed. The data and calculations are shown to be in good overall agreement and consistent with the predicted shell-model ground state configurations and the one-neutron overlaps with low-lying states in $^{18-21}$C. These suggest significant $\nu{s}_{1/2}^2$ valence neutron configurations in both $^{20}$C and $^{22}$C. The results for $^{22}$C strongly support the picture of $^{22}$C as a two-neutron halo nucleus with a dominant $\nu{s}_{1/2}^2$ ground state configuration.