Sadly, yes. They can also get bladder infections (cystitis). However, it's not as common in horses because of their size. The symptoms are pretty standard - increased frequency of urination with less fluid being produced each time, and sometimes cloudy or bloody urine. And while horses don't get them as often, they can be much more serious in horses being much more likely to progress to cystitis or, worse, kidney infection. As in other species, mares are more likely to get them than stallions or geldings. Also, horses with urinary tract infections often have signs of infection elsewhere, especially on the skin.
It's treated with antibiotics or antifungals (horses can, yes, get "yeast" (candida) infection).