Main Scenario Quests never directly require particular side-quests to be completed, but they do all have a minimum level, and if meeting that level requirement is the issue, then side-quests are one of the options for getting a bit more XP.

From your Lodestone profile, it appears you've at least gotten up through the Ul'dahn Envoy quest that unlocked airships. (That's actually a fairly common place for people to lose track of the Main Scenario, if they get side-tracked into exploring the new areas and content and then forget where their last Main Scenario quest left off by the time they want to continue with it.) That Envoy quest should have been followed by quests sending you into a series of three group dungeons, first Sastasha Seagrot, then the Tam-Tara Deepcroft, then Copperbell Mines. Did you do those? Anyway, you can check the "Completed" tab of your quest journal against this list of level 15 to 20 quests:

The Ul'dahn Envoy (level 15)
Call of the Sea (level 15)
It's Probably Pirates (level 15)
Call of the Forest (level 15)
Fire in the Gloom (level 16)
Call of the Desert (level 16)
Into a Copper Hell (level 17)
The Scions of the Seventh Dawn (level 17)
A Wild Rose by Any Other Name (level 17)
Unsolved Mystery (level 17)
What Poor People Think (level 18)
A Proper Burial (level 18)
For the Children (level 19)
Amalj'aa Wrong Places (level 19)
Dressed to Deceive (level 19)
Life, Materia and Everything (level 19)
Lord of the Inferno (level 20)

If you've completed all of those except for Lord of the Inferno, then it is a level issue, as that's the first Main Scenario quest to require level 20. In that case, just level up a bit and you should be able to continue. Good ways to level up are by repeating the dungeons you've unlocked (particularly by doing Duty Roulette which gives an extra daily bonus), doing side-quests, completing your Hunting Log, or doing FATEs.

If you're missing some of the others in that list, then it's not a level issue. You just need to find the next quest. The list above is the order they come in, so find the last one you've completed, and the next one listed is the one you need to do. The links go to GamerEscape's wiki pages for each quest, and that wiki page will list the Issuing NPC and where you'll find them.