Is it correct to use a comma both before AND after a conjunction?

There is often talk about when to use a comma before conjunctions, but what about when to use them before and after? I would typically write the below sentence as follows:

There is a weight of expectation on her, but, judging by her recent performance in
class, she has a great shot at getting into Harvard.

I have seen many things online that say you really should not be using commas after "but" and only before when it starts an independent clause. However, there are those times when you need to use them both before and after.

It does make the sentence look choppy when you use a comma before and afterwards, but it is technically correct comma usage from what I can see? Would it be perfectly correct to just omit the second comma in order to create a smoother sentence:

There is a weight of expectation on her, but judging by her recent performance in
class, she has a great shot at getting into Harvard.

Which is correct?