The difference is almost always in just knowledge of the game. If you know more about the champions that were picked and the compositions that were formed than your opponents, you will always have an edge over them. This is one thing you should never stop improving on.

You should also understand that poking is not necessary to winning lane. In some cases, overcommitting to landing poke may actually cost you in the long run. You should always try to diversify your champion pool so that being on pokey champs isn't the only thing you are comfortable with. Generally, the hardest snowballing supports are Blitzcrank and Thresh, both of which are very poor at poke but extremely good at winning lanes when played well.

There is also passive play that can result in winning lanes from overcommitting opponents and poor decision making on their parts. Champions like Janna or Braum who are generally poor at poking but excel in peeling are great choices for making your opponent feel like they need to make something happen and then punish them extremely well when they try.

And of course there are many other things you may want to improve on, such as warding. But in general I find that the easiest way to carry on support is to be a good shotcaller. This means telling your team when to back off from things, especially if they're about to respawn or whatever, telling your ADC when to play more careful because you're out of wards, or when to take dragons or when to do or bait barons. The ability to shotcall is probably the hardest but most awarding ability to learn as a support or even a jungler if you play that role, as it allows your team to have a grander strategy than none at all, which is very helpful in the randomness of soloq.

So tl;dr: learn more champions, probably Blitzcrank, and learn to shotcall well. Also ward more.