Puddin' I'd like to suggest you do a google on the words "counter steering". All two wheel vehicles use this inbuilt trait to both balance and steer the bike. It may well be that your instinctual skill and higher level car driving skills are at odds with each other and this is why you're having trouble steering the bike.

Altering the style of bike to get different geometries doesn't change the fact that if your steering skills are confounded by conflicting messages in your brain that you'll end up riding in a hesitant and wobbly manner.

Now this is very likely to start a HUGE argument because this is an extremely hot topic but I'm pretty sure from what you're describing your riding skills to be like that if you read about and then go and practice the DELIBERATE and CONCIOUS use of counter steering that it'll make you a better and more able rider. But the key is to conciously push the right grip to go right and the left grip to go left. This includes when you're balanced in a turn. If you're carving around in a right turn you actually push the left grip to go left and straighten up the bike. Once you're doing it conciously with good control you'll find that your brain will file this new reactions into the instictive bank and it'll come more naturally to you.

You see, riding a bicycle is sort of like balancing a broom handle in your palm. You need to keep getting your palm back under the broom handle if it starts to tip. And if you want to move the broom handle and yourself from "here" to "there" you need to actually make the handle tip towards the way you want to move to get it started. Then you follow the handle until it's where you want to be. Then to get it back in balance you push your hand ahead of the falling handle to get it back into balance.

The bike works the same way. You actually tip the bike to start a fall on purpose then you steer into the turn to balance it. Finally you steer FURTHER into the turn to lift the bike back up to vertical.

Each and every turn is broken down into these same three phases as when you're balancing that broom handle. A)The deliberate roll into the lean by pushing on the grip towards the desired turn. B)Countering this deliberate roll by turning INTO the fall. And C) the recovery where you overcorrect for the turn by turning even further into the turn to move the bike back under yourself and roll it back to vertical by pushing on the grip on the outside of the turn. And yes it is all backwards to how you steer a car. And that's why it can often give some folks troubles with conflicting messages to the muscles if you try to rely on instincts and early childhood experience combined with your car driving experience.

If all this sounds like Greek then I'd suggest you need to practice it and play with it in a safe area. Don't take my word for it at face value but read what I've typed along with looking at the other google results including some videos about this and then go and conciously try applying it yourself. I know that if you approach it with an open mind and let the bike show you that it'll be a "EUREKA" moment. I'd be very surprised if you don't end up being a more capable rider with better bike control skills as a result.

I'm going to go and get my Nomex flame proof underwear on now because I know this whole topic verges on religious belief with some. I just ask that you read about it and then go and try it using deliberate and concious bar control consistent with counter steering principles. I'll let the bike speak for itself.