We present results of three wide-band directed searches for continuous gravitational waves from 15 young supernova remnants in the first half of the third Advanced LIGO and Virgo observing run. We use three search pipelines with distinct signal models and methods of identifying noise artifacts. Without ephemerides of these sources, the searches are conducted over a frequency band spanning from 10~Hz to 2~kHz. We find no evidence of continuous gravitational radiation from these sources. We set upper limits on the intrinsic signal strain at 95% confidence level in sample sub-bands, estimate the sensitivity in the full band, and derive the corresponding constraints on the fiducial neutron star ellipticity and $r$-mode amplitude. The best 95% confidence constraints placed on the signal strain are $7.7times 10^{-26}$ and $7.8times 10^{-26}$ near 200~Hz for the supernova remnants G39.2--0.3 and G65.7+1.2, respectively. The most stringent constraints on the ellipticity and $r$-mode amplitude reach $lesssim 10^{-7}$ and $ lesssim 10^{-5}$, respectively, at frequencies above $sim 400$~Hz for the closest supernova remnant G266.2--1.2/Vela Jr.
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