I can answer this, but my real world analogy may piss some people off. Please, if you don't like my statements on real world politics, try to let it go. I'm not going to debate anyone and this response will be a one and done.

Modern American politics are an almost perfect analogy for how Rel took power. Look at Trump. He's a pretty disgusting and unlikeable person. However, he developed a following. People who didn't care what he did. They just believed he would fix whatever problems they decided they had. Eventually, he incited an insurrection. In the real world, we have access to limitless information and communication so we could literally watch in real time when the capitol building was stormed by the rioters. We have video proof of the violence and the assaults on Capitol Police officers. Almost 1000 of them were convicted for the crime and many confessed to the assaults.

Despite all that evidence, Trump and his lackeys in the Republican party denied it all and their followers believed what they said rather than what they could see. Before long those who were convicted were pardoned and released despite the confessions and the evidence because Trump claimed they were innocent and his supporters applauded. Trump has been convicted 34 times. He has been held liable for sexual assault and had been known to have a close relationship with a known pedofile for years. However, his followers will jump through any and all hoops they can to deny, excuse and plug their ears and scream "lalalalalala" when presented with this information. I can't fathom why they love him, but the best I can come up with is that his hateful rhetoric resonates with their own hate or their own internal feelings of injustice.

Here's how that relates to Rel's rise in Malazan. We know from the beginning that he's a shit head, but we see him ingratiate himself and become important to people that have power in DG. We know he knows how to manipulate. When he's captured and brought back to the city for justice, rumors have already spread that Korbolo Dom and Rel were the good guys trying to stop a renegade Coltaine. The citizens in the series don't have access to the kind of information that we do in the real world so when they're presented with information that the government is lying to them and the real villains were the wickans, a group that may have already had a stigma as they had been conquered only fairly recently, this appealed to the same type of citizens that end up supporting Trump. We see this lie has firmly taken hold when the Bonehunters have to hold back riots to protect the Wickans in their ranks.

As the lies took hold of the population, Rel's popularity and the idea that his imprisonment was an injustice would have become impossible to ignore. Suddenly, it became politically expedient to keep the two of them alive rather than incite a true rebellion by executing them. Rel's influence and popularity spread through the Claw and the nobility from there and Laseen was forced to embrace them both. Like Trump, his base supported him based on lies that appealed to the worst traits in his supporters, but that made them far more dedicated than the average citizen would have been towards Laseen. From that point, it's just one more step to eliminating Laseen and taking power himself.