Is it necessary to perform istinja' (washing the private parts) and wipe the back passage with tissue for every prayer in a case where flatulence is frequent and one often does not feel the exit of anything impure?

It is not recommended to perform istinja' (washing the private parts) after passing wind. In fact, some scholars have considered it an innovation. Scholars unanimously agree that istinja' is not obligatory after passing wind, sleeping, touching women, or touching the male organ. If you are certain that something other than wind has exited the anus, then istinja' is obligatory. If you are not certain, then you are not required to do it. Wiping the area is not necessary merely based on delusion or doubt. If the emission of wind is continuous, such that it does not stop for a period allowing you to purify yourself and pray, then you take the ruling of someone with incontinence. You perform wudu' for each prayer after its time has begun, and you pray even if wind exits from you, and you are not required to repeat your wudu'.