He fails most likely.
Dragons are a huge force multiplier and make conquest relatively easy but securing a throne is an entirely different matter. Essos spent millenia under the aegis of the Dragon Lords and once the Doom occured the Essosi were not shy about assassinating any who escaped the cataclysm.
The Targaryens would need an army large enough to actually secure Essos and make it safe for trade, which means wiping out the Dothraki. They would need a population that at least acquiesced to being ruled by the Dragons. And they would face constant assassination attempts. In addition, Essos was (and is) a land of isolated city states without any feudal tradition.
Westeros was a very different situation. The Lords of the Seven Kingdoms could not abandon their castles else they would lose the allegiance of their lesser lords and small folk, and yet to refuse to bend the knee was to see the dragons burn those castles to ash with no real risk.
Then you had the religious angle. If needs be, the Targaryens could make common cause with the First Men and in doing so solve the single largest factor preventing them from wiping out the Andals; castles and large armies of knights. The Faith could bend the knee and acknowledge Aegon as the rightful King or they could declare full blown religious war and Aegon would burn Old Town to ash along with any other lord or city that sided with the Faith while an army of First Men floods South.
In addition, Westeros had a tradition of feudalism and only five truly relevant nobles (Gardner, Lannister, Durrandon, Arryn, and Stark). If they bent the knee then with them would come the vast majority of their vassals; and if they refused then finding a hungry vassal with a "legitimate" claim willing to support the Targaryens in exchange for the recognition of their claim was never going to be any issue.
In the case of Westeros, all Aegon had to do was show up with a relatively small army, declare himself king, give a demonstration of what air power meant to the balance of power in Westeros, and he essentially had a thrown nominally secured.
At which point keeping it was always going to be relatively easy as you still had the competing power bases of the various great lords and they could always be counted on to betray one anothers plots to the Throne as they feared one another gaining the Throne more than they hated the Dragons. Not to mention that Aegon had already proven that he was perfectly willing to burn castles and armies and everyone knew that if you plotted against him and failed than he would just take his dragon over your lands and burn you and your supporters to ash without you able to resist.
More, he also had a stick to hold over the Essosi. They could acknowledge his claim to Westeros and business could continue as usual (better actually as Westeros became a better trading partner) or they could plot against him and risk a dragon deciding that they were just more trouble than they were worth and the Black Dread makes a short trip to start setting every population center on the continent aflame.
The mistakes that Aegon made were many, but his basic target selection wasn't one of them.
