We present Data Releases 4 and 5 of the quasar catalog from the quasar survey by the Large Sky Area Multi-object Fiber Spectroscopic Telescope (LAMOST), which includes quasars observed between 2015-September and 2017-June. There are a total of 19253 quasars identified by visual inspections of the spectra. Among them, 11458 were independently discovered by LAMOST, in which 3296 were reported by the SDSS DR12 and DR14 quasar catalog after our survey began, while the remaining 8162 are new discoveries of LAMOST. We provide the emission line measurements for H{alpha}, H{beta}, MgII, and/or CIV for 18100 quasars. Since LAMOST does not have absolute flux calibration information, we obtain the monochromatic continuum luminosities by fitting the SDSS photometric data using the quasar spectra, and then estimate the black hole masses. The catalog and spectra for these quasars are available online. This is the third installment in the series of LAMOST quasar surveys that has released spectra for ~43000 quasars to date. There are 24772 independently discovered quasars, 17128 of which are newly discovered. In addition to this great supplement to the new quasar discoveries, LAMOST has also provided a large database (overlapped with SDSS) for investigating quasar spectral variability and discovering unusual quasars, including changing-look quasars, with ongoing and upcoming large surveys.