By rules as written, you can only grapple creatures one size larger than yourself (PHB p. 195), so a medium creature can jump on a large and a small onto a medium.
Note that the second paragraph is an alternative, not a variant, so it too is in play. If the creature is larger than you can grapple and is suitably large, whatever the DM decides that is, you can simply use your movement to walk or climb it. The DM would need to decide which and if the creature was difficult terrain or not and if any checks are needed; presumably Dexterity (Acrobatics). Assuming the creature didn't want you there it could try to throw/pull/buck you off, resolve this as an opposed check.
Personally, I think it would be awesome if the players jumped and climbed up the dragon to try and sever its spine. They would too; right up to the point where I say "The dragon drops prone and rolls over you. Time for some saving throws."
As RAW, no, it does not
Being considered a size higher for carrying capacity and push, drag, or lift force is not the same as being a size higher for all purposes, including grappling.
You count as one size larger when determining your carrying capacity
and the weight you can push, drag, or lift.
As RAI, maybe
The loop hole is that the PHB has no reference on why and how the grapple-er can move without speed penalty if it is two size higher, and the DMG does not touch the subject. There are two potential scenarios for why a grapple-er one size or lower has it speed reduced, though; the grappled creature have enough leverage to make the movement difficult or that the grapple-er cannot carry comfortable enough the grappled creature as in "difficult terrain", and that the weight of the creature is the deterrent.
In the first case, since Powerful Build does not increase per-se the size, it is safe to assume that it does not help to increase the comfort or reduce the leverage and, such, the speed is halved.
In the second case, where is the weight that maters, things change. Powerful Build improves the carrying, dragging, and lift capacity, therefore the grapple-er creature does not have its speed halved.
Which one it is
The only evidence I found is in favor of is that weight is the reason for the reduction in speed. The first part is in the text of moving a grappled creature (emphasis mine).
When you move, you can drag or carry the grappled creature with you,
but your speed is halved, unless the creature is two or more sizes
smaller than you.
As far as I can tell, there are no rules on how you drag or, particularly, carry a grappled creature. You can carry over you head a grapple creature as if it were a sack of potatoes. In PHB 176 there is a segment that shows how can you drag, carry and lift something in particular and, as shown, it is a STR and weight contest in which Powerful Build should work as intended.
Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up
to twice your carrying capacity (or 30 times your Strength score).
Bottom line
At the end is the DM that has the last word. Talk to your DM and present the evidence. If you are the DM decide what it is best for your campaign. If a particular decision is more powerful that you expected it is in your right to take back the decision.
Best Answer
Yes, it is possible, Tasha's introduces the Rune Knight, and 'Giant Might' the third level feature allows you to grow to Large size, furthermore 'Runic Juggernaut' at level 18 increases that size to Huge. All you need then is an enlarge spell and you're good to go!
Yes, I know this is 2 years old (almost three) but I still wanted to respond :)