But without some objective basis for arguing that something is wrong its impossible. All you would be doing is arguing that your personal opinion should be taken as fact. Thats normally not how truths/facts are determined. Of course not. But that does not negate there are moral truths. The moral arguement doesn't say that atheists don't know whats right and wrong. It only questions what the grounding is for the morals being claim to be wrong under subjective morality.
The point is we make child abuse an objective wrong and thats our measurement. You don't need to be religious to do that. We all intuitively know child abuse is wrong. It would be counter intuitive to say "child abuse is morally ok". We would think anyone who thought it was OK was objectively wrong and there was something wrong with them.
Actually I see it the opposite way around. Under a subjective moral system moral behaviour cannot improve or evolve into anything because its just about feelings, preferences, opinions and they cannot be right or wrong.
But appealing to improvement and progression of morals is also appealing to an objective measurement that can tell us when things improve or even find better/best ways to act morally. I found this articles explanation of how subjective morality is hard to apply and doesn't really fit with how we treat moral matters.
No Room for Social Reform and Progress
One of the strongest objections to relativism is the idea that if relativism is true, then there can be no such thing as social reform or moral progress. If each culture’s ethical code is equally good and right, then when a country changes its ethical code from being pro-slavery to being anti-slavery this moral change is merely a change rather an improvement. Moral improvement and progress require that there be some standard toward which a society or an ethical code are approaching; they also entail that the subsequent morality is better than the prior morality, but again this is not something that can be said if relativism is true.
When the United States abolished slavery and segregation, and gave women and minorities the right to vote, its ethical code underwent a change. But to say that it underwent an improvement requires saying that enslaving African Americans, segregating Whites from Blacks, and preventing women and minorities from voting are objectively worse, morally speaking, than their opposites. Relativism cannot consistently support such a position for relativism entails precisely the opposite, namely that there are no objective standards for morality and morality is relative to communities. If a community decides that it wants to endorse X and then later decides to morally condemn X, then both moralities are equal. No morality is superior to another.
Aren’t Right and Wrong Just Matters of Opinion? On Moral Relativism and Subjectivism – Introduction to Philosophy: Ethics
I am trying. I covered Ethics 101 such as deontology, teleology, Utalitarianism, Consequentialism, Kant and all that. Studied Mills, Ross, Parfit some others. Most articles regardless of moral position will cover arguements for and against each position i.e, moral antirealism. But its a big topic with lots of sub topics so I will keep reading and learning.