Praise be to Allah.
Death and lifespans are determined by the will and decree of Allah, may He be exalted, that He wrote in al-Lawh al-Mahfooz with Him fifty thousand years before He created the universe, and it cannot be changed or altered, for it is decreed by His knowledge that does not err and by His will that does not change. Allah, may He be glorified and exalted, says (interpretation of the meaning):
“And spend [in the way of Allah] from what We have provided you before death approaches one of you and he says, ‘My Lord, if only You would delay me for a brief term so I would give charity and be among the righteous.’
But never will Allah delay a soul when its time has come. And Allah is Acquainted with what you do” [al-Munaafiqoon 63:10-11]
“He said, ‘O my people, indeed I am to you a clear warner
[Saying], Worship Allah , fear Him and obey me
Allah will forgive you of your sins and delay you for a specified term. Indeed, the time [set by] Allah , when it comes, will not be delayed, if you only knew’” [Nooh 72:2-4].
All of that does not mean that death and lifespans are not subject to the laws of cause and effect that Allah has created in this universe. Rather the issue of death, like everything else that is decreed in this world, is based on material causes and effects that were also written in al-Lawh al-Mahfooz.
What a person should do is be moderate in all his affairs, and in his estimate of causes that could lead to his death. So he should not exaggerate in being careful and taking precautions, because if he does that and gives free reign to negative thoughts, he will think that every movement, and perhaps every moment of stillness, could lead to his death, so simply going out of his house and walking in the midst of cars, trucks and so on may be a cause of death. But thinking in this way is wrong, and no rational person would think that way.
At the same time, he should not fall short in taking appropriate measures to be safe. So he should not expose himself to danger or neglect to take appropriate measures of being safe, whilst believing that all things are in the hand of Allah, and that he is only taking these measures in obedience to the command of Allah, may He be exalted, then after that whatever Allah wills will happen.
We would like to remind the questioner of a brilliant example in which there is a lesson, from the life story of the great Sahaabi, Sayyiduna Khaalid ibn al-Waleed (may Allah be pleased with him). He spent most of his life fighting on the battlefield, then he died in his bed. That is because Allah, may He be exalted, decreed that he should die like that.
Hence we know that the fear of buying a motorbike may be for a valid reason, if the questioner or anyone else thinks it most likely that it is more risky or more likely to lead to accidents than other means of transportation, either for a specific individual whose driving skills are lacking, or because motorbikes are more prone to accidents, as is the case in many places, or for similar acceptable reasons. There is nothing wrong with all of that, and avoiding riding a motorbike in such cases is a type of taking precautions as prescribed in Islamic teachings, and avoiding things that could lead to one’s death, whilst believing in the prior decree of Allah, and that Allah has already set for everything a [decreed] extent.
What is not right in that regard is when the fear of riding a motorbike, or doing anything else, arises for no apparent reason, and with no rational explanation. In that case, it is a type of irrational fear or phobia, so you need to strive to overcome it and rid yourself of it, even if that is by means of acceptable psychological treatment, always keeping in mind the belief in the prior decree of Allah, and that taking measures does not change the divine decree at all; rather that is part of what Allah has decreed.
And Allah knows best.