First of all, let me say... this is completely normal. I was just as bad and all new moms face this problem. Knowing how crazy things can get, you don't want to be caught out somewhere unprepared (wait till she potty trains :)
I'm a huge fan of ziploc bags. Wipes stay moist you can carry less and you can see through the bag to know if you need to refill. Just bring small amounts of things and tuck them into her bag and put it in the stroller. If you don't have a small bag to put them in, look at thrift stores or Kmart/Walmart/Marshalls/TJMaxx for something cheap. Not taking a stroller? Chances are if you're carrying your child it's not the purse that's killing your back. If your child is walking, you aren't going out for very long anyway because kids don't last that long, so you don't need to bring all that much stuff.
When my kids could drink from a straw, I was so happy. Sippy cups were the worst to carry and clean out. The straw was the best improvement in our lives.
Think about all the things you are carrying. If you are carrying a whole container of wipes, that may be heavy, but other than that, her stuff isn't weighing that much. Try pulling out all the things for her and weighing them. I will bet it is no more than 1-2lbs. That's nothing to carry. If you're carrying a lot for you in that bag too, maybe you should consider what you really need and simplify (most likely you need little more than a wallet, keys, cell). If you are carrying your daughter, maybe you should consider using a stroller more often. I loved strollers, then I didn't have to carry anything (and I tucked in a full container of the Wet Ones wipes, because I had room). I switched to a tiny wallet that I can slip into my pocket. I put that in one pocket and my cell in another pocket and then I don't have to worry about guarding the stroller contents as closely.
You haven't even mentioned toys which I find interesting. If you're using a stroller, you can clip the toys on to the side so you don't loose them and there's nothing to carry. Make sure your stroller has a bag on the back or a compartment underneath. If not, consider investing in a bag you hang on the back or just slip the handles of her little bag on the handle.
I used to carry SO much more stuff when I had a stroller because it was basically free space and I could. Then when I tried to minimize the stroller usage I found myself being smart about what was really needed. I threw 2 packages of fruit snacks in my purse (2 kids) and made them wait to drink when they got back to the car. (Easier to do with a 2-3 year old.)
Another thought is to invest in a backpack style purse/bag. I got a black leather (or pleather, I don't remember) one that looked nice but allowed me to distribute the weight across both shoulders. I would use that when I was Christmas shopping because I wanted the stroller to hold my purchases and I could fit extra toys inside the bag to switch out so the kids would last longer shopping. I've never paid more than $30 for a purse, even Liz Claiborne so I probably got it at a discount store.
Don't worry, it is a short phase of her life. When my youngest was about 3 I challenged myself to carry just keys, cell and wallet so I didn't have to keep up with the purse. I wanted to have hands free to hold my children's hands. We went to the car if we needed a pull-up and we washed hands in bathrooms. I always kept a spare set of essentials in the car (wipes, pull-ups, change of clothes).
Good luck J.. Happy Veterans day. Thank you for supporting our military and please thank your husband for his service.
-Liz