Perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, it should be noted that when someone travels to commit a sin, it is impermissible for him to use the concessions granted to travelers of combining and shortening the prayers according to the majority of the scholars. If he is traveling for a lawful purpose, then he may use those concessions.
As for the person suffering from incontinence, if he is unable to wash his private parts and perform ablution for every prayer, he is granted a concession to combine the prayers according to the scholars who deemed it permissible for such a person to combine the two prayers that may be combined; so he may combine the Thuhr with the ʻAsr and the Maghrib with the ʻIshaa’. In this case, it would not be difficult that the bus stops in a place where he can wash his private parts, perform ablution and pray. If all this is difficult for him, he is exempted from the conditions of prayer that he is unable to fulfil and is obliged to fulfil what he is able to fulfil of them. So, if he is unable to remove the impurity, he is exempted from this obligation and he may perform the prayer in that condition.
It should be noted that the Maaliki scholars held that it is not obligatory on a person suffering from incontinence to remove the impurity if he discharges urine involuntarily, even if it happens once a day. They also held that such a person is not required to perform ablution for every prayer if he discharges urine (or any other impurity) half of the time or more. There is no harm on the person who faces hardship in cleaning himself properly to act upon this view. For more benefit, please refer to fatwa 330445.
As for facing the Qiblah, it is one of the conditions of prayer without which the prayer is invalid. So it is impermissible for him to perform the obligatory prayer on the moving bus because he would not be able to face the Qiblah as required. Rather, he is required to try to stop the bus (i.e. ask the driver to stop) and perform the prayer on the ground or inside the bus while it is parked. If this is not possible and he cannot perform the prayer on the ground or in the parked bus even if he combines prayers, then he may pray in the state which he is in. He should try to face the Qiblah to the best of his ability and fulfil what he can of the conditions and pillars of the prayer, and he is exempted from what he is unable to fulfil.
Allah knows best.