Hiya, OP, you've got three things going on here I think. You will have the same issue in any car seat, but don't worry, it's all fixable.
First of all the coat. Although it's very cold, it's actually not recommended to wear coats in the car for the reason you are finding. They add so much bulk, that even if you can do them up, they make the car seat straps much too loose can cause positioning issues like this where you see that the shoulder straps are nowhere near her shoulders, which would be very dangerous in a crash, as she would likely not be held in the seat. It also looks uncomfortable for her.
So if you take the coat off, then obviously she'll be cold. You can remedy this in a few different ways.
Look for a thinner coat - if budget allows, you could look for a coat which is made with a thinner material like a fleece, or boiled wool, or a soft neoprene type material. Appreciate this might not be a realistic solution, especially as coats tend to be expensive this time of year. I did see some nice thinner ones in Decathlon recently. If the material of the coat is not so puffy, then you can put the straps on over the coat and it will still run nice and close and snug to her body which is what you need. With a puffy coat like this, you can't compress the coat enough to get the straps close to the body.
If you can't get a thinner coat, look at different layers - if you have a fleecey top try wearing this under the coat to walk to the car, and then take the coat off at the last minute. A warm vest under her other clothes will also help, a hat, and gloves.
You can put coat back on, over the straps, backwards - so her arms are toasty and the coat is kind of like a blanket. Older children usually find this fun - 2 year olds might get annoyed and kick it off. So you could also bring a fluffy or fleecey blanket to the car and tuck that in over her once she's strapped in. If you have a blanket you don't mind cutting up, you can make a poncho type blanket by cutting a hole for her head, which again allows straps to be done up underneath and is difficult to kick off. You can often find brightly coloured or character type blankets cheaply, which appeal to young children.
If you have the option to stick the engine on and start the heaters up 5-10 mins before you need to be in the car, that can work (appreciate not everybody has private parking).
An option to combine with any of the others, fill a hot water bottle or use a microwave rice heating bag just before you leave for her to hold onto in the car on her lap, which will warm her hands and her core. It doesn't usually take too long for the heating to kick in and this can help get over the initial cold hump.
Also distraction - do a silly dance in the car while waiting for the heating to kick in, sing a loud song together, play a silly game to distract her. Some people get all the kids inside the car with coats on while the heating warms up, then they get out, coats off and strap kids in then get back in and drive.
The second problem is that I think you've got the headrest set too low for her here. In the Joie seats, as the child grows, you can move the headrest up and this allows you more space in the straps. You want to look for a little squeeze button or lever on the back of the headrest at the top. If you squeeze this you can move it up and down. You want to have it so that the headrest is level with her head and her shoulders fit underneath it. This will instantly make her more comfortable and so the straps aren't halfway down her arms any more. It should also give you a little bit more length to work with (but you'll find it much much easier in any case once you ditch the puffy coat). I can't see if you have any inserts in but if you do you can likely take those out as well.
Lastly just a minor issue, but your strap is twisted in the buckle, it will be much easier to use if you slide the buckle up or down the strap until it's on a flat part of the strap and that twist is gone. If it ends up pointing the wrong way around, fold the strap, and push the buckle over the fold so that it's facing the right way. To avoid twisting in the future (Joie seats are notorious for it) completely lengthen the straps when you take your child out and lay them over the sides of the seat. It's digging under their bum that makes it twist, but it's easy enough to fix again if it does happen.
Good luck - let me know if you need any further info!