During the 1980's, the American Strategic Defense Initiative studied using laser weapons to defend against ICBM's. Since the powerplant, cooling system, optical trains etc. are large and massive, launching laser weapons in space was considered to be a non viable option, but a massive ground installation only could strike targets in line of sight.
This led to the consideration of "fighting mirrors", which would be launched into Low Earth Orbit (or in extremity, on a ballistic trajectory), which would reflect the beam to the target over the horizon from the ground installation. A secondary benefit was the mirror could possibly be used to refocus the beam once it passed through the atmosphere, bringing it back into sharp enough focus to damage or destroy targets.
SDI Mirror concept
APOD experiment. While not optical quality, this mirror gives you an idea of what the system might look like deployed.
Now the laser was meant to strike missiles in space, and most illustrations only envision one or two laser mirrors. Given the massive amount of laser energy, the lack of cooling systems on the lightweight mirrors and speed of engagement, I suspect the mirrors were meant to be sacrificed with each shot, since even absorbing a tiny percentage of the laser energy would likely destroy the mirror. Even if the mirror wasn't sacrificial, it seems prudent to rotate the laser to different mirror sets so each mirror has time to cool and retarget.
So the system you envision was designed a long time ago, and works mostly as you think, except that the laser beam is transmitted outside the atmosphere to strike targets in space. Attempting to strike ground targets would require the beam to pass through the atmosphere again, leading to issues like thermal blooming and other optical degradation of the beam, so this does not seem to have been considered at the time.
The key element is to minimize the number of times the beam has to be "handed off", since it will lose energy each time it is reflected, and to only pass through the atmosphere once. Indeed, if the laser installation is powerful enough, the beam might trigger the atmosphere to break down into a plasma, which absorbs the beam (and runs down the beam like a lightning bolt to the optical train), so there are definite boundaries to laser energy as well.