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We present the results of the light sterile neutrino search from the second Karlsruhe Tritium Neutrino (KATRIN) measurement campaign in 2019. Approaching nominal activity, 3.76×106 tritium β-electrons are analyzed in an energy window extending down to 40 eV below the tritium end point at E0=18.57 keV. We consider the 3ν+1 framework with three active and one sterile neutrino flavors. The analysis is sensitive to a fourth mass eigenstate m42≲1600 eV2 and active-to-sterile mixing |Ue4|2≳6×10-3. As no sterile-neutrino signal was observed, we provide improved exclusion contours on m42 and |Ue4|2 at 95% C.L. Our results supersede the limits from the Mainz and Troitsk experiments. Furthermore, we are able to exclude the large Δm412 solutions of the reactor antineutrino and gallium anomalies to a great extent. The latter has recently been reaffirmed by the BEST Collaboration and could be explained by a sterile neutrino with large mixing. While the remaining solutions at small Δm412 are mostly excluded by short-baseline reactor experiments, KATRIN is the only ongoing laboratory experiment to be sensitive to relevant solutions at large Δm412 through a robust spectral shape analysis.