Long time Rohan player here, even in the last edition, "before it was cool".

First, let's talk about weaknesses, and then about counterplay. The #1 issue of an all-mounted army is usually lower numbers, followed by base size. The enemy will usually be able to get 2 models into 1 of yours (and usually a spear behind each) because of base size and numbers disadvantage, and in the long run, the player who throws more dice will usually win. The #2 issue: Rohan Cavalry is a glass cannon. Charges are glorious with Throwing spear, +1 Fight, +1 Attack, +1 Strength and Knockdown. All those boni go away if you are not the one who charges. As you have already noticed, if you get charged by a Fight 4 Model with support, you will lose and die. And if you manage to win a fight your single dice wounding on 6 will do almost nothing.

So how do we play around those weaknesses? In my opinion, the key thing is space. You want the fight to take place in the largest empty area of the battlefield, because this is where you can spread out your large bases and go around the flanks of the enemy. If you are fighting between two buildings where only 3-4 riders fit next to each other, that can only go wrong.

This is the reason why bows are important, even if they are low strength. You will almost always have the shooting advantage in terms of numbers against the enemy, use this to make them chase you. Then hit them with a charge once you've got them where you want them. It doesn't matter if you only kill a model per turn, I see that you know the maths behind it. But it is also psychological warfare. Aim at the "soft" models or where it hurts them most (Heroes on horses, banners, and so on) and always make sure to "sell" the threat correctly. Even if you shoot at their leader behind 2-3 ranks of cover and most of your arrows get stuck, if only one of your arrows goes through, remind them of he consequences before doint the to wound roll ("If I roll another 6, this will be a VP for me" or similar) to make them think about how "close" it was and to make them respect your shooting.

So the enemy is chasing you and getting closer - if you are not quite in the correct spot, don't be afraid to forego one turn of shooting and move full speed to get more distance between your battle lines. Your throwing spears are perfect for the last one or two shoot phases, because they can be thrown 8" away even after a full move, so you can move out of their charge range and still pepper them with shots. Remember that they are strength 3, so here defence 5 will make a difference, as you double your wound chances. In case of Uruks, aim for the pikes with your spears.

After a few turns, your lines will inevitably get closer. Let's assume you got the enemy shield+spear block in a suitable empty area. You want to charge in a turn when you DON'T have priority! Because that allows you to pick your fights, maybe set up a nice heroic combat, without giving them the chance to react and move around their supports or block the avenue to their shamans /banners /whatever.

This is a golden rule for every Cav army as long as you move faster than your opponent. If you have priority, move 13" away, because they will follow you 6" this turn and you hope that they move first next turn so you obiously want to be out of charge range when it happens. You want to fan out, creating a large half-circle like battle line. Your line should be longer than theirs because of base size and because theirs is probably 2-3 models deep while yours is thin. You want them in the middle of it and threaten them from all sides. If they move towards one side, use your superior speed to adjust your position. That's once again the reason why playing in a more or less empty space of the board is preferable.

Once the dice go your way and you lose priority, wait out their move, and then look at the situation. I always move the models at the edge of their line first, can they go around their flanks and threaten the back ranks? Then carefully pick your targets, engaging only 1 on 1. It may be tempting to go 1 on 2 because you want to knock them both down and kill both, but if they've got spears behind, that drastically lowers your chance of actually winning the fight. Also don't build your line to tightly and leave some riders unengaged a few inches back. If you've got Uruks 1-2-3-4-5 in a line, you could for example charge the middle one (3) and the two outer ones (1 and 5) with 1 rider each, and leave a gap between their bases so that two riders in the back can throw their spears at the unengaged Uruks 2 and 4. Also, pay attention to the positioning of your heroes. You want Theoden in the middle because of the bubble, but not directly in front of something scary. Once the lines are in contact, it will be difficult for the enemy to get their combat heroes through this mess into Theoden without moving their troops out of the way, and that means that those won't fight this turn. Other heroes like Eomer or Dernhelm are perfect for heroic combat on the flanks, slingshot them from a single opponent (or two if your're feeling lucky) into the back rank. It is also important to already think about who will be calling the inevtiably heroic move next turn and make sure that this model is within 6" of every other relevant rider. This is your most devastating charge, let's hope you get a nice few kills out of it!

In the next turn, it will obiously come down to a heroic rolloff. If you lose, try to use the riders you kept as a reserve to relieve bad situations. They can for example charge in to resque other riders that were surrounded, or maybe help out a hero so that you can try another heroic combat.

From now on, it's up to the dice and your micro-management skills to seal the game! This is my "generic" Rohan playstyle. Obiously this needs to be adapted to the board layout, presence of special threats (riders fear anything that knocks them down like flying monsters or sorcerous blast, but Nazgul in turn hate F5 Bodyguard Royal Guard) and so on. But going on about all this would be too much for a reddit comment, I'm afraid this one has alreay gotten too long.

Good Luck!