How does Renji become stronger than Rukia? I only have the anime to go on so maybe there's an explanation in the manga. It seems clear that Ichigo is stronger than Rukia was in episode 1 at the point of his first battle with Renji. In fact it seems like Ichigo may have been stronger as soon as he became a soul reaper, but based on Rukia's reaction to the stronger enemies up to that point (Menos Grande, and Grand Fisher) it seems she believes that a single shinigami could hope to win in cases where Ichigo does. This seems to imply that Ichigo is stronger than Rukia ever was at that point. Then Ichigo fights Renji and is at least closely matched. It's difficult to say that at the point where Ichigo gains the upper hand whether he is stronger or weaker than when he was fighting stronger enemies in the past but it does seem like Ichigo and Renji, at their first meeting, are at roughly the same strength which is greater than Rukia's at the beginning of the series.

But in episode 32 where Renji has his flashback to his time in the Rukon district and training to be a shinigami it seems clear that Rukia was always stronger than he was. So then how do we explain this discrepancy? Did Renji get significantly stronger in the few months Rukia was in the world of the living? Is there any particular reason for that? Or am I underestimating how strong Rukia was at the start?
When Rukia stepped into the world of the living, she had to inhabit a "Gigai" (empty body) to be able to interact with the living. That body was provided by Kisuke Urahara. Hidden inside that Gigai was one of his inventions (the Hōgyoku), which by being placed in the same body as a Shinigami would be destroyed, or so he planned. After Rukia gifting her Shinigami power to Ichigo, her powers failed to return, and the Hōgyoku was what was blocking Rikia's powers. Along with the tower she was held in during her wait for execution, she was left with no power. Shortly after she was saved, the Hōgyoku was stolen, and her power began to very slowly return.

In the later episodes, when she has recovered, you can clearly see her strength after she has enough power to call forth her Zanpakutō with its overwhelming ability to freeze enemies.

Just as Ichigo has gotten stronger from numerous fights with Hollow and Shinigami, so has Renji from their fights and his own determination to save Rukia from her execution and later protect her from further harm. Also, his power was boosted considerably with him mastering his Bankai at the same time Ichigo was unlocking his own.

Plus, all Shinigami have their power capped/suppressed when visiting the human world, or they would seriously damage the living.