I bought my Escort with 85,000 miles on it, with four tyres that were obviously 'used' but still with a reasonable amount of tread on them. It's now got 140,000 miles on it (so that's 55,000 miles I've driven, starting with part-worn tyres). I always replace tyres before they get below (or even fairly close to) the legal limit and, in all, I think that I've had just one full set of tyres in that time. (There's still loads of tread left on the ones currently on my car).
Over the years (with various cars), I've generally worked on the assumption that a set of tyres will last at least 30,000 miles and I've often got closer to 40,000 miles out of them. (I generally buy the cheapest tyres available, as long as the grip rating is reasonable; I ignore things like the noise rating).
So 13,000 miles to me sounds pathetic! (I've bought part-worn tyres in the past, when I've been hard-up and I'd have expected at least 15,000 miles from them!).
Are you sure that you're checking your tyre pressures regularly? (Indeed, are you sure that you're using the correct pressures? Most cars I've had required higher pressures in the rear tyres than the front but my Escort requires 35psi in the front but only 29psi in the rear).