No, fire is not inherently stronger or weaker than the other elements. There's a considerable amount of variation in strength and skill among benders, with some able to pull off feats beyond others. For example, Azula can lightningbend and generate blue fire, which are beyond Zuko's abilities. On the other hand, Zuko does learn lightning redirection, which Azula lacks. On average, however, each element is equally strong insofar as it is possible to assign a strength to an element.

The more important consideration is what you want to use the element for. Fire is the most inherently destructive element, which makes it particularly useful when attempting to kill people. To add to this, firebenders can simply conjure fire from raw chi, in contast to earth- and waterbenders, who need an external supply of their element to hand. (Airbenders also need an external supply of their element, but that only becomes a problem if they're in space, where the other elements aren't present either).

Beyond this, the Fire Nation has a huge military which is exceedingly well-equipped. You know those steam-powered metal ships and tanks? The Water Tribes and Earth Kingdom lack that technology, which gives the Fire Nation armies an advantage.

Another factor, shown in season 3, is that the Fire Nation is fairly centralised and makes extensive use of propaganda. This gives them a highly devoted and patriotic population who are entirely willing to join the army and fight with zeal, as they believe that the world is rightfully theirs, and that the other people will be better off under Fire Nation rule.

Oh, and the Fire Nation hasn't conquered most of the world. At the point you're at, it's effectively neutralised the Southern Water Tribe and rules about a sixth of the Earth Kingdom. However, most of the Earth Kingdom is still free, as is the entire Nothern Water Tribe.