It's difficult to recommend a seat without being able to see your car and set up. You might look to see whether you have any independent car seat specialists near to you, or John Lewis tend to be good out of the chains if there are no non-chain stores.
First I'd suggest to look at your current seat and see if there are any adjustments which can be made. For example, do you have newborn inserts in the seat? Check the manual to see when they should be removed, but for infant carriers, often newborn inserts should stay in until around 3-4 months or roughly 60cm. At 15 weeks, your daughter might still need the insert or it might be pushing her up and out of the seat and need to be removed. For fixed seats which stay in the car and last several years, the newborn insert or infant insert is often intended for use up to around 9-12 months and may have multiple parts to remove as your baby grows. Again double check in the seat manual, which you can usually download online if you don't have a paper copy.
Secondly, it's likely at 15 weeks that she needs the straps raising up from the lowest position, if this hasn't been done yet. Having the straps set too low will push her into a more curled up position where she might fall forwards. You want to aim for them to be just level with her shoulders or slightly below (no more than 2cm lower). Most infant car seats you move the straps up by moving the headrest up and down, but in some seats it is a different mechanism perhaps accessed from the back of the seat.
Third, some babies at this age are starting to want to sit up and look around more. This can mean that when you put them into the car seat, rather than resting back in a lying position with their shoulders in contact with the seat, they are reaching forwards and looking at what you're doing with the seat buckle for example. This tends to be more common in seats which are fixed in the car and cover a longer age range. But if this happens, when you tighten the straps they can't then self adjust which tends to mean they have a gap between their shoulders and the seat which makes them fall forwards. Try to use the seat in its most reclined position and lay baby down between your forearms and place her in a lying position into the seat (as you would for a nappy change for example). This is also where it helps to have the straps level or close to level with her shoulders as then they will tighten nicely to keep her in that laid back position.
Fourth the problem might actually be the angle of the seats in your car. Many cars have sloped back rear seats, which causes a problem with baby seats which are rear facing. If this is the problem and your seat is on an isofix base, check whether the support leg is not overextended and if so, shorten it until the seat base is touching your vehicle seat but the leg indicators are still green. Then check whether the base has any tilt adjustment options as some do. If it does not, you might get a better result by removing the base and seatbelt fitting the car seat, if this is possible for your model. With seatbelt you have a little more flexibility of angle and can often tilt the seat slightly further back. Do check the manual to be sure it's OK to fit the seat like this or go back to your retailer so they can check the fit for you.
If your car has sloping seats and low down isofix, you'll likely have a similar problem with most car seats. In order to counteract that, you can look at seats which accommodate a rear seat angle such as the Britax or Joie seat bases with tilt/angle adjust, or for the next stage seats you could look at Besafe izi Twist/Turn or Axkid One, though these seats are expensive so I'd absolutely recommend visiting a stockist to try them out in person before buying (esp as your daughter is on the cusp of whether she will need the baby inserts, which add £100 to the price of each). Belt fitted seats might also work well for the next stage e.g. Avionaut Sky, but she is too little for some of the cheaper options such as Britax Safe Way M.
Or some infant carrier seats can recline when installed in the car. Cybex seats can do this and the Cloud G is on a good deal at the moment on amazon.
If the problem is not sloping seats but you still don't like the fit you get in your current seat after making adjustments, I would probably look at the Cybex seat as mentioned or Avionaut infant carriers (Cosmo or Pixel) because these have really good inserts which support younger babies well. Or if you prefer to move on to a longer lasting seat, I'd probably look at Avionaut Sky/Stardust or possibly Joie i-Pivot, though a lot of people struggle with the Joie inserts, so I would recommend (again) trying it out before purchase if possible.