Is it possible to go from E-4 to E-5 and directly go to Drill Sergeant school?

Not in my day. You need a level of maturity that a newly promoted E-5 doesn’t usually have. If they allow it now, I would be very surprised. You need to have some leadership time first. A specialist doesn’t.

Branch (assuming they are paying attention at all) is not going to send a brand new, inexperienced E5 to Drill School. The whole point of Drill Sergeants is to train future Soldiers. A brand new E5 doesn't even know how to how to be an NCO yet, much less have the experience and knowledge needed to be a Drill Sergeant.
That's how you end up on the cover of Army Times...

Well considering the importance of a Drill, you wouldn't want to, and they wouldn't want you if you're a fresh SGT. You need experience as a NCO before you're ready to be Drill.

In the reserves, yes. You have to be promotable and be granted a conditional promotion, complete the school, and then you are official.

For males you have to be an E-6 unfortunately from what I’ve been told. I wish to be a DS as well. Speak to your chain of command and let them know your goals. I was given great advice, “always think two grades ahead.” For you that would be PFC/ SPC. Obviously have goals. Work towards those goals. But make sure you’re hitting your short term. Get that expert marksman badge. 270+ pt score above 500 ACFT score. As a 25B go get your A+ Net + Sec+. Get some college under your belt. Get your correspondence courses flowing. You’ll get where you want to be by doing the little things others don’t want to do.

It is possible in the reserves. You had to have completed BLC and be an E4 promotable before attending the course. However, I’ve seen other replies to this post that I tend to agree with. You should first have some leadership time as a sergeant. You should also be proficient in PRT and SMCT level 1 training. A Drill Sergeant has to be able to lead, train and mentor civilians straight off of the street and transform them into soldiers. Some of them have to be taught very basic things, like how to shave properly, or how to properly apply sunscreen. Some of them have never done laundry. It’s a profession in which you wear many hats. Make sure you’re ready before taking that leap.

PFC (Join to see) Slow Down there Lad. Get a Little Experience Under Your Belt First. To Share Experience You have to Have Some. Authority is Great but You Have to Earn it First.

Technically possible, but highly unlikely. As a new SGT, you hardly have any experience has a leader or a trainer. Being a Drill Sergeant was one of my favorite assignments. Why are you in such a hurry to be a DS?

Once upon a time that was possible and was often done in the mid sixties. In those days we were so short of personel that many thing were done that are not normlly done in peacetime. However they were sort of picky about who they sent to the school.

Why do you want to become a Drill Sergeant?
To your question, it is possible, but unlikely. Once you are a SGT, WLC graduate, and have 4 years of service, you can apply if you meet the remainder of the requirements.
You're a few decades too late, t one point they had Drill Corporals.