(Note that some of the things in this thread are my just personal thoughts and not an objective truth about game design.)
According to the Plus documentation, this is what 'Turbo' difficulty is.
As it stands right now, the Turbo difficulty feels a little undercooked. It fulfils its purpose of being "harder than 'Hard'" but because of its current implementation, there's no real incentive of picking 'Turbo' instead of 'Hard'. I think it has a lot of potential and to explain what I'm trying to say, I'll go over each of the gameplay changes it currently brings.

Max health is only three hearts
This is the change about which I have no strong feelings one way or the other. On the one hand, this makes the player become more careful of their surroundings and makes damage boosting a riskier move. On the other hand, it incentivizes backtracking for carrots to make traversing the rest of the level safer. Take this as you will but I think this is leaning more towards being a good change rather than a bad one.

Buttstomping does not give you invincibility
I have nothing to argue against this change, disabling this mechanic prevents cheesing the game, especially for some of the bosses. If 'Turbo' gets a rework, I think this should be kept in it.

Enemies are harder to kill and may exhibit unique behaviors
This is a good change but... they do? The "enemies are harder to kill" part of it is somewhat obvious if your weapons are not powered up, they do have more health. But had I not read the Plus docs, I wouldn't have known about the enemies' different behaviors in 'Turbo' difficulty. The changes are very minute, I had to play a bit on 'Turbo' after reading this just to confirm to myself that yes, there are in fact some differences. My suggestion would be to amp up their new behaviors so they're both noticeable and fun. Here are some examples of what I have in mind:
- Dragons could shoot flames further instead of faster, the flames could be denser.
- Hatters could shoot three projectiles instead of one, similarly to upgraded RFs.
- Fencers could keep fencing towards the player continuously after being triggered to do so once (so long as the fencer enemy is on screen).
And so on. The same enemy redesign philosophy could be (and I believe it should be) applied to bosses as well. Maybe the Queen wouldn't let her guard down if the player is too close to her. Or maybe Schwartzenguard's flail would home in on the player for a little while. You get the gist of it.

Pressing the fire key repeatedly does not allow you to shoot more frequently than holding the key down
This change bothers me the most. As it is right now, it's both pointless and annoying. Pointless because fastfires can still be obtained and max fire speed is easily obtainable in two or so levels. Annoying because it takes some control away from the player and makes the start of the game rather sluggish. Its purpose is clear: Make it more difficult to kill enemies. With this in mind, I have two ideas that I believe would improve the player experience while not making the game easier:
A. Remove the manual fire cap and disable fastfire drops from enemies
This solution keeps the player in control of their fire rate and removes the insane buff that Blaster + Max Fastfire. However, there are two problem with this: the player might be tempted to create macros to spam the Shoot key or they spam the Shoot key so much that they develop carpal tunnel. Because of this, I thought of another possibility.
B. Keep the manual fire cap to a medium value and disable fastfire drops from enemies
This way, the player can shoot at an adequate fire rate chosen by the developer while completely eliminating the possibility of shooting at a crazy high fire rate. This change should apply to weapons that don't have autofire as well for the same reason; fast homing missiles and RFs are devastating to bosses.
For both of these ideas, the fastfire pickup should still work the same. If it was the level maker's intent to have the player get them, that should be respected.

Other potential changes
These ideas aren't related to the already implemented gameplay changes.
Remove carrot drops
Simple as. I believe most level makers don't rely on the player to get random health drops from enemies, that's why most SP levels have static health carrot pick ups for the player to get. Removing these random drops would make the game a bit more difficult while not going against the level maker's wishes.
Different menu sprite color palette
This would be a nice way to differentiate between Hard and Turbo. I believe a grayscale version of the sprites should look pretty cool. I don't know about the palette limitations of the game menu however, so if this isn't possible, well, bummer.

Well, I think that's about it. If you liked - or disliked - my ideas, please vote in the poll in this forum post. New ideas, tweaks to my ideas or any kind of interest in a potential 'Turbo' difficulty rework would be greatly appreciated as well. Thanks for reading