There is a growing concern about an impact of low-Earth-orbit (LEO) satellite constellations on ground-based astronomical observations, in particular, on wide-field surveys in the optical and infrared. The Zwicky Transient Facility (ZTF), thanks to the large field of view of its camera, provides an ideal setup to study the effects of LEO megaconstellations-such as SpaceX's Starlink-on astronomical surveys. Here, we analyze the archival ZTF observations collected between 2019 November and 2021 September and find 5301 satellite streaks that can be attributed to Starlink satellites. We find that the number of affected images is increasing with time as SpaceX deploys more satellites. Twilight observations are particularly affected-a fraction of streaked images taken during twilight has increased from less than 0.5% in late 2019 to 18% in 2021 August. We estimate that once the size of the Starlink constellation reaches 10000, essentially all ZTF images taken during twilight may be affected. However, despite the increase in satellite streaks observed during the analyzed period, the current science operations of ZTF are not yet strongly affected. We also find that redesigning Starlink satellites (by installing visors intended to block sunlight from reaching the satellite antennas to prevent reflection) reduces their brightness by a factor of 4.6+/-0.1 with respect to the original design in g, r, and i bands.