Hi,
during TUR either BT (or P = Prostate - Guys only!) a burn or direct cut is effected. This tends to bleed to a greater or lesser extent after surgery, which is why, both to ensure minimum pressure in the bladder pulling on the wound (however minor) and also to constantly wash the bladder to remove any free floating particles, blood and maintain a non acidic environment to aid healing a catheter is left in place and the output bag is checked for blood colorant.
Sometimes a large amount of fluid will be run into and out of the bladder through a two way catheter and it may take a couple of days before bleeding stops sufficiently to remove the catheter.
Anything from a week to a month later, again tending to be dependent on the amount of resection done the bladder interior surface will have healed adequately and the 'scab' that formed will float off - this may be passed in bits over a couple of days or sometimes as a single mass. Just as it is never right to pee claret and even rose should be taken seriously - although worrying peeing brown bits after an op or a grey/brown caterpillar is somewhat disconcerting but panic not it is just the scab.
As a simple guide - red colours are live or recent blood but brown or greyish tends to be long dead blood from a scab or similar. IF this lasts for more than a day or two take it seriously and contact your urology specvialist or Doctor.
You sound as if you have nothing to worry about even if it is hard to convince yourself that The Big C stands for Challenge and Choice and TUR in less medical terms is a Trans Urithral Reprieve
Good luck and remember it is YOUR Bladder and YOU Choose when YOU need reassurance - that said never ever ever miss or delay a check up or treatment.