Stability

I agree it depends on what you were used to driving and how you were driving it.
The Jimny is a short wheel base rear wheel drive car which handles completely differently to a Front wheel drive "ordinary" car.
The suspension is set up for off road work so is a bit stiffer than standard cars.
Although it has coil springs the axles are rigid and not independent as on cars so you get some reaction from bumps.
Generally it's not made for "quick" driving and is quite happy around 55 ish on the road.
Tyres generally last a long time so go hard with age so you'll find new tyres can make a difference but it'll never be like a car.
Before you go out and get new tyres just check the tyre pressures as it wouldn't be the first time that garages have put the pressures too high.
It's 23 front 26 rear on standard tyres.

I have a 03 jimny with lift kit and other off road mods and your right it's not a super smooth ride on the road but awesome off road which is what the car's suspension was designed for as long as your daughter doesn't go breaking any land speed records and cornering at silly speeds it will should be a very good and reliable car for her with the added advantage of a higher seating position and also when the snow comes she should get home safe and sound, I've had mine about 12 months and prefer to use it than my 10 plate honda civic it's just more fun and fits through gaps that only bikes and pedestrians can follow !!
Im running at tyres for general use which are 195 80 15 but I'm guessing standard size will be best for insurance purposes.

As above, check the tyre pressures.

That sounds much like how every Jimny drives. Tall, narrow, short wheelbase, high unsprung mass, unsophisticated suspension. It's a recipe for a choppy ride! But there might be some small things you can do to improve it...
As Gilburton says, check your tyre pressures first!
Then look closely at the tyres. If they look old and perished, or have the word 'Dueller' written on them, replace them with any branded modern tyre (that isn't a Dueller) and you will notice a difference (once you've let the pressures down again, because the tyre fitter will insist on putting them at 35psi...).

As far as first cars go, I'm sorry but I'd say the Jimny is a poor choice. The on-road performance, safety, comfort and economy is simply miles behind modern small cars. Unless your daughter likes playing in the mud, she'd be better off in an early 90s Micra!

Hi all and thanks for the advice, tyre pressures were at 30f and 36r now set to 23 f and 26r already better on the road,
took a look at the tyres they look perished on the side walls so will replace them in the very near future, the reason we went for a jimny is we live up the side of a mountain in sunny Ireland, I run an l200 wife Suzuki vitara, eldest daughter started with a micra great car, until the snow comes, will let you know how she is handling once the tyres are changed thanks again to all of you

Glad you found a reason for at least some of your problems.
Given your location a Jimny is probably a good car to choose and most roads are rural with mud etc.

Once I took my car to a Suzuki dealer for a service and when I got it back I thought they had removed the springs.
Checked tyre pressures and they were at 35 all round.
They must have given that job to the "lad" lol

More years ago than I care to remember I was a tyre fitter. If we weren't sure of the pressures we put in 25 psi as that was the average of tyres then.
These days low profile tyres are fitted and the suspension is "tuned" to take the road shocks and it stiffens up the handling at speed.
Because of this tyre pressures are now way above 30 psi as the tyre takes little part in absorbing bumps.

The Jimny is an old design and has large "balloon" type tyres (as do other 4x4 cars)
The pressures are therefore old school as well but they just pump in the average all the same!!