The rest-frame far-ultraviolet morphologies of eight nearby interacting and starburst galaxies (Arp 269, M 82, Mrk 8, NGC 520, NGC 1068, NGC 3079, NGC 3310, and NGC 7673) are compared with 54 galaxies at z ~ 1.5 and 46 galaxies at z ~ 4 observed in the Great Observatories Origins Deep Survey (GOODS) taken with the Advanced Camera for Surveys onboard the Hubble Space Telescope. The nearby sample is artificially redshifted to z ~ 1.5 and 4 by applying luminosity and size scaling. We compare the simulated galaxy morphologies to real z ~ 1.5 and 4 UV-bright galaxy morphologies. We calculate the Gini coefficient (G), the second-order moment of the brightest 20% of the galaxy's flux (M 20), and the Sérsic index (n). We explore the use of nonparametric methods with two-dimensional profile fitting and find the combination of M 20 with n an efficient method to classify galaxies as having merger, exponential disk, or bulge-like morphologies. When classified according to G and M 20 20/30% of real/simulated galaxies at z ~ 1.5 and 37/12% at z ~ 4 have bulge-like morphologies. The rest have merger-like or intermediate distributions. Alternatively, when classified according to the Sérsic index, 70% of the z ~ 1.5 and z ~ 4 real galaxies are exponential disks or bulge-like with n>0.8, and ~ 30% of the real galaxies are classified as mergers. The artificially redshifted galaxies have n values with ~ 35% bulge or exponential at z ~ 1.5 and 4. Therefore, ~ 20%-30% of Lyman-break galaxies have structures similar to local starburst mergers, and may be driven by similar processes. We assume merger-like or clumpy star-forming galaxies in the GOODS field have morphological structure with values n < 0.8 and M 20> – 1.7. We conclude that Mrk 8, NGC 3079, and NGC 7673 have structures similar to those of merger-like and clumpy star-forming galaxies observed at z ~ 1.5 and 4.