The simple answer to your question is no - infant car seats which have a carry handle (known as "group 0" seats) cannot be turned around and used forward facing. A popular example is the Maxi Cosi Cabriofix.
You can get a car seat which does turn around and go forward facing, this is known as a combination seat or group 0/1 seat. An example of this is the Britax First Class.
You can also buy car seats which allow for rear facing for much longer than the usual group 0 seats, these cover groups 0, 1 and 2 or sometimes just group 1 and 2. These are called Extended Rear facing seats or ERF.
If you want a car seat that clips into a buggy, then you are looking for a Group 0 seat. Seats which accommodate older children are just too big and bulky to clip into a pushchair to make a travel system. And yes, this means that you're only using the seat for a short time (although the time frame is more like 12-15 months unless you have a HUGE baby in which case they might outgrow it by 9 months.) Advice which states 6-9 months is outdated and refers to older seats.
If you're stretched on budget, I would advise looking for a deal with a pushchair and Group 0 seat together, as you save money over buying the Group 0 seat separately. You don't need an isofix base if you're trying to cut costs. Then save up for a decent Group 1 seat - high backed boosters, which are the next stage up, are very readily available in good quality brands from about £30-40 if you're careful and watch the deals.
Group 1 seats tend to be sold "from 9 months - 4 years" but really this isn't the case - group 0 seats are bigger than they seem. The maximum weight for a baby in a group 0 seat is 13kg which can be reached anywhere from about 1 year of age, for a huge baby, to 2.5 years of age for a very small baby. They usually outgrow the seat in height before weight, but because this is measured by the top of the head, again, most babies are between 12 and 15 months when they hit this point. DS was small and he fit into his until he was 18 months old, by which point I could no longer carry the seat with him in it! Many people think that you have to put babies forward facing by 9 months, but no - as long as they fit into the seat by length or by weight then they are fine and safe to continue using the seat.
And remember, if you plan to have more children, you can reuse the seat because the first will be at least 9 months old by the time their sibling is born even in the most extreme scenario! :) So that first seat is more of a long term investment.