Is it true that dogs lose trail of a running person if this person crosses a bank or a small river?

NO!! That's useless.
A good tracking dog will CAST....pick up the scent trail where it comes OUT .

&YES,water will...DOES...carry scent!
Cadaver-dogs can &do locate corpses underwater! From above.

Going through water will stop a trail, but most good tracking dogs will go across the water, unless the river is too big? and pick up the trail on the opposite bank. It is possible to confuse tracking dogs by going up or downstream for a distance, before coming out on the other side.

Are you planning a 'great escape'? LOL

Bank, as in the sides of a stream or river no - it always has to be water to stop a direct trail.

they would lose the persons sent once they entered the water, so they would then cross the river and try to pick up the sent on the other side.
so if you were running from a dog that was hunting you, you should walk through the river. that way they won't be able to pick up your sent.

sorry, i should add that if the river is very small they will be able to smell the track on the other side of the river before they even cross it. but it is definitely impossible to smell a sent on the water.

a teacher once told us that a dog scent is so strong, that if you put a 1/2 cc of a drop of blood in a gallon of water the dog would know that there was blood in the water. I went to college for animal health, the dog's nose is amazing, but it's for sure to say whether they can or can't lose the scent.

They would lose the trail in the water, however they'd be able to pick it back up once the person got to the other side of the running water.

They can't track through running water because the water carries the sent away.

No. The scent would still be on the water, however it would be hard to track it. Unless it is moving water, then yes. You would need to get to the other side to resume tracking.

NO