Riding and handling a pony can be quite dangerous. I know you know that, but I've seen lots of accidents happen to children in similar sorts of situations to you. I wasn't someone completely born into horses, but I worked as a riding instructor for a while, so I am sympathetic, but I also don't want your daughter hurt or put off by a nasty fall, or the pony hurt.
I do think you need someone experienced supervising your daughter's riding- for her sake and the pony's. At 7, she's too young to deal with a situation like this on her own, and it's impossible to tell what's to be happening, over the internet.
As a first step, before you ride the pony again- have a read/look at basic saddle fit and common problems with saddles- a badly fitting saddle often causes bucking. Ponies do change shape with the time of year so something that fit last summer might be causing issues now.
To check the fit, you should put the saddle on the pony without a numnah/saddle pad. Looking from behind the pony, you should be able to see daylight through the gullet, the saddle should look balanced and not skewed if the pony is standing square. Check for any pinching around the shoulders or the withers- and check there is a decent clearance (about 4 fingers) over the withers. Check the saddle is not coming down the pony's back past the last ribs. Put a girth on the saddle, and get someone to trot the pony up. Check the saddle is not moving or bouncing on the pony's back.
These are only very basic checks, but may help highlight a problem with the saddle fit. Alternatively, when you get an instructor out, ask them to check the fit.
You should also check the underneath of the saddle for any damage. Feel the flocked (padded) areas for any lumps/bumps, and check the tree is not poking through uncomfortably near the front. It's a good idea to check the saddle regularly anyway, just in case any stitching has broken, or there is any damage.
If you think there are any issues with the saddle, talk to the owner, and try not to be fobbed off.
Find someone who can lunge for you- even if it's just an enthusiastic 16 year old (an instructor would be better)! Get them to lunge the pony without the saddle, and with the saddle but no rider. See if the pony behaves the same in both these situations.
Does the pony have any areas of sensitivity when you're grooming? Are they girthy? If you run your hand (firmly) down the back, do they show any signs of pain?