Not sure what may have caused your drop, not that experienced myself as I just bought my first bike back in May. So I can only say a few things I do that seem to give me confidence in starting out.
When I first head out, I generally baby the **** out of the bike till I make it to a big road. I want the tires warm before I give it much throttle at all. So pulling out of my garage and drive way I am barely giving any throttle in turns, and often am sliding my motorcycle boots along the pavement just in case. Once I am on the road and pointed straight I will give it a little gas, but I do slow down more than normal when making the next two turns to get to the main road. I have to pull out into traffic at that point so I wait for a big gap. I pull slowly onto the road, and wait till I am pointed straight. At that point I gun it and drive my normal wreckless self.
Probably not the right way to do it, but I will say I haven't even been close to dumping it with this method (knock on wood).
I have ridden bicycles hard for decades, and basically I do the same thing on a bicycle. I have dumped my bicycle way too many times trying to peddle hard in a turn to start out.