While it can be difficult to come up with an in-universe explanation for something that has an out-of-universe explanation of different effects techniques, and possibly a much better budget, let's try.
TL;DR - We can theorize that the more regenerations a Time Lord has gone through, the more difficulty they have during/immediately after a regeneration.
Looking back to the regenerations from the original series, we see no significant mental trauma or confusion involved in the earliest regenerations. While the 2nd Doctor was different from the first, most of the confusion in his regeneration came from his companions, who simply weren't entirely sure this was the same person. That element didn't really come up again, from what I've seen (he had active companions during the regenerations from 3->4, 4->5, 5->6, 6->7, 9->10, and 11->12, at least).
Starting with the 4th regeneration, into the 5th Doctor, we started to see significant mental issues - calling people by the names of old companions, and the like. While the regeneration into the 5th Doctor seemed the worst (and the Master may have had something to do with that), the 6th and 7th Doctors both showed some confusion and distress at first (haven't watched the 8th Doctor movie recently enough to recall - but he did say he was part human then, which has basically been disavowed by now, I believe).
Something unique to the 4->5 regeneration was the Watcher - who was, apparently, the Doctor in a state in between his 4th and 5th forms, bouncing back in time to "watch" the 4th's last adventure. This can be handwaved away for any of a number of reasons - his visit to Logopolis and its unique nature, and the possibility of the Master's involvement in order to manipulate him into going to Castrovalva being the two that leap to mind.
Starting with the War Doctor, we've seen a certain amount of physical trauma as well, in the form of the regeneration fire apparently causing the physical change. I haven't seen the regeneration into the 12th Doctor yet, but the explanation that the majority of the change was accomplished earlier, and the "sneeze" just adjusted the cosmetics, seems roughly credible to me. (I think we were also told that the Doctor had been putting the full regeneration off for some time, leaving his 11th form to limp along after it should have been replaced, which could account for some special oddities here.)
(That said, the idea that the regeneration from 9 to 10 wasn't using the fire as a standard regeneration technique, but as something caused by the Vortex energy, also seems entirely credible out-of-universe. And, so does the idea that, whether due to the team involved forgetting that fact, or simply due to the "coolness" factor, this became a part of the standard regeneration).
This leads to a logical (though not explicitly stated in canon) theory that the older a Time Lord is (or the more regenerations they've gone through), the harder a regeneration is on them.
I should note that I believe there are two regenerations that happened off-screen: the regeneration from the 2nd Doctor to the 3rd, and the War Doctor to the 9th.