Well you could do it I suppose but it might be a bit of a waste of money. At 4 your dd wouldn't get so much a lesson as a lead rein ride with some fun things thrown in, e.g round the world, touch the pony's ears, etc. When she is able to balance in walk, trot and canter she would probably enjoy a childrens group lesson. Once you can both ride to a good standard you could enjoy hacks together.
OP - I think if she wants to have a go now is the ideal time. She'd be able to have lessons on the lead rein and they need not be too expensive - especially if your local riding school does group lessons on a weekend morning.
Your lessons would need to be at a different time though as you'd be on horses and probably for longer lessons. Eventually though you would be able to go hacking/trekking together if you had somebody else to lead her or you felt confident enough to lead off your own horse.
A lot of people start their children as early as 4 (and even earlier), but in my opinion most kids can't get much out of lessons until they are about 6/7. This doesn't mean that they won't enjoy the experience but it all depends on what you want your child to get out of it.
I think 4 years old is too young tbh. To get the most of it, you need a certain level of resilience as there's nothing surer than you will fall off at some stage! I think most 4 year olds would be put off by a fall. I learned to ride when I was seven years old and it was fine then.
In your position, I wouldn't try to learn at the same time as your DD as your needs will be quite different. It might help if you have some lessons first, then you would be able to support her when she learns.
I learnt to ride at 4. One of my early memories is falling off and seeing the sky spin round before hitting the ground lol! Didn't put me off though. Little kids learn fast and often have very little fear. Most riding schools will take them from 4 and I think it's a good age to start. I agree with other posters you'll probably want seperate lessons though as you'll be taught in very different ways.
My friend runs the pony club at a centre in the midlands. She has a lovely tiny tots group. I've helped out loads. It's great fun and the kids love it. Yes they do occasionally fall off, but. I've never seen one who hasnt got back on. I think the trick is to find somewhere that doesn't over horse little ones. Shetlands or welsh As are plenty big enough. It would be nice to start learning with her, if you are both total beginners then you can do all the exercises too, but she would probably also benefit from a group lesson, as she would make friends, and it's good to have friends to compete with. I'm not sure you would necessarily fit into a four year olds lesson!
I have ridden off and on for 20 years. DD began lessons when she was 6. A lot off private ones to get her up to speed soon and when she was 8.5 yrs we started doing a group class togther. It is lovely mum and daughter time. We have on holday done a private lesson for us to. I tend to canter more than her, but it works fine and we are doing something together which I love as only have 5 weeks annual leave. I didn't feel she had the physical strength (although tall for her age) before she was 6 years old, but then again she didn't start out on a shetland.
I think that if you can afford it, it's a great idea. Of course at 4, DD will be doing basic things (pottering around on a leading rein and gentle fun stuff as described above - lovely for 4y.os, though!), but she'll still get a feel for horses and riding, which if she takes to them, will be really valuable to her. I started riding at 4 - loved it, still do. CeeYou is right, though, you and she won't fit in in the same classes, and she'll be on the lead rein for a bit longer than you will (we hope!) - you might be able to do some gentle hacking together, though, and you could watch each other's classes, maybe?