That's why pumps were invented, but good try OP.
NO! No they, don't, they work on gravity and atmpospheric pressure. The whole point of a siphon is that the outlet is lower than the inlet. Yes, the water may move to a higher level during the siphon, but the outlet needs to be lower. You can't cheat physics here... Siphons will stop immediately if you bring the outlet up to a level near the inlet. In a best case scenario you could get several inches out of a column of water due to atmospheric pressure (read: water barometer).
If you make this loop work I would apply for a patent and publish a paper to submit for a Nobel Prize. Unless you start adding in dynamic stages like evaporative towers or something you won't get this result out of consistently liquid-water.
OP, I'm not sure if you've read this
at all or not, but you need to have air in the lines, because it cheats and uses a less dense fluid (air) to push up which allows it to be driven by the falling water. Also worth noting is that it is NOT a perpetual motion system, or even close. You're looking at a few minutes unless you have a constant feed source (which would defeat the physics because you'd have to have a costant sink too). That said, you may well be the one to succeed where Bernoulli, Euler and Newton, along with countless others weren't, so by all means, give it a shot, just make sure you know your physics first.