How Can God be Real if There is so Much Suffering and Evil in the World?
There's certainly a great deal of injustice and sadness and suffering in the world. Nobody can deny that. As for how this affects the existence of God, that very much depends on what one thinks God is in the first place. It's easy to think of God as some sort of "overseer" or king figure who created the world a long time ago and then watched it develop and then when He sees fit, intervenes here and there to fix things. This is basically what children are taught to believe, and we aren't taught much more in adulthood.
But the above idea or definition of God is not correct. It certainly is not the idea of God in mainstream Sunni Islam nor is it the idea of God in Ismaili Islam. In proper Islamic thought God is the "source of existence" and God exists beyond time and space and change. This means that God's creative activity is eternal and continuous and He is sustaining the existence of all created beings at every instant. This also means it's incorrect to think of God as a personal being with feelings and wishes who sometimes intervenes in the world - because such a being would be temporal not eternal. And any being that is subject to time is created. What is laid out here about GOD's existence can be proven using logical deductive arguments as found here: https://ismailignosis.com/2014/03/27/he-who-is-above-all-else-the-strongest-argument-for-the-existence-of-god/
This has bearing on the problem of human suffering and evil. This is because an eternal God who always sustains existence is never withholding blessings and favours from His creation. Through His single and eternal act, God eternally bestows and pours out His loving compassion (rahmah), power (qudrah), life (hayah), knowledge (‘ilm), favour (ni‘mah), grace (fadl), forgiveness (maghirah), inspiration (ta’yid), and illumination (nur), etc. upon all created things. This is confirmed by numerous Qur’anic verses as follows:
Of the favours of thy Lord We bestow freely on all – These as well as those: The bounties of thy Lord are not withheld.
– Holy Quran 17:20
That which God opens unto mankind of mercy (rahmah) none can withhold it; and that which He withholds none can release thereafter. He is the Mighty, the Wise.
– Holy Qur’an 35:2
My Mercy (rahmah) encompasses all things.
– Holy Qur’an 7:156
Instead what happens is that different types of creatures possess different capacities to receive God's blessings. Atoms and molecules and plants and animals each receive from God what their capacities and limits allow them to get. It's the same for humans - we receive from God what our hearts and souls can allow us to capture.
Human souls are therefore varied and differ with respect to their receptivity to the resplendent lights of the Divine Command, just as material objects are variously receptive to the physical light of the sun. [Consider] stones, for example: one [kind] is pitch black, while others are progressively less dark, and their essences are more receptive to illumination, up to translucent glass which receives light from one side and emits from the other.
– Sayyidna Nasir al-Din al-Tusi,
(The Paradise of Submission, tr. S.J. Badakhchani, 109)
But humans are more advanced beings who, using intellect and intentional action, can actually modify and change their own hearts and souls for better or worse - to receive more or receive less. The outcome depends on our own thoughts and deeds. Because of this human capacity to better ourselves we also have the ability to harm ourselves and others. Being human is a double edged sword - we can become like angels or become worse than beasts. God endowed humans with this capacity but He is not morally responsible for a person's bad or evil choices. That's upon each person. While God is responsible for the possibility of evil in the world, He isn't responsible for evil as such - humans and rational creatures are. More about this idea of God always bestowing blessings and how it relates to prayer is here:
https://ismailignosis.com/2015/03/22/ya-ali-the-rationale-for-praying-to-god-and-calling-upon-the-imams-in-prayer/
Finally, the question itself is proving that God exists because it implies that there should be justice, good, compassion, and mercy in the world, and that there should be no suffering. Whenever you're able to look at the world and recognize evil, bad, and suffering, it's only because you first recognize the values that are good and true and beautiful and compassionate. Evil is the absence of these values. So this begs the question: do these values like goodness and justice and truth and compassion exist objectively and outside of human minds that think about them? Does the good exist independent of whether all humans recognize and understand the good to be? If God does not exist and only the physical world exists, then all these values of goodness and Justice and compassion are just ideas that human species constructed for social and evolutionary purposes. These values, without God, have no independent objective existence of their own; at best they exist in the intellect of a person who holds them and believes in them, and they cease to exist when the person no longer thinks them or affirms them. This means that without God or a level of eternal spiritual reality beyond the physical world of time and space, there really is no such thing as "good" or "evil" in any objective sense of these words; instead good and evil are just social constructs that differ from each person or whatever culture or society deems to be right and wrong. The ONLY way for moral and ethical values like goodness, justice, and compassion to exist in a true objective sense - independent of human opinion - and physical existence - is if these values always exist beyond space and time in an eternal intellect which is always thinking these values: this eternal intellect is the Mind of God. If objective values like good and evil exist, then God exists.