One of the things to remember as braum is your wave clear is crap, so your options are sunfire cap or a ZZrot portal for split pushing
As a support you should always have a method to aid in a split push
ZZ rot is usually the better option but if you find yourself constantly stopping now giant minion waves because your team is ignoring them go sunfire
In lower Elo I would get both, knowing how to split push as a support gives you a much higher chance of success because you rely less on your allies for it
As an example I was support poppy once and my team was losing badly every fight, I got a ZZ rot portal (didn't need sunfire because poppy has aoe, just like tahm won't need a sunfire, braum would though) and I would set minions up to split push then drop the ZZrot where it would reach there tower and returned to my team for the team fight
This took out 4 towers for my team, unfortunately our adc played like he was a front line tank and the rest of my team wasn't much better, resulting in our loss.
But an important thing to keep in mind as a support is minion control, that doesn't always mean killing all thier minions, leave 2-3 to make your minion wave grow and force them to stop it or lose a tower
If they go to stop it it will be a 5v4 in your favor if they don't you get a tower
There are obviously other things to do as a support, but that's one way to carry the game as a support, but some are better at it and at times it's better to just group