Learn Turkish: İki rakip şirketi uzlaştırmak çok zor bir iş. - Reconciling two rival companies is a very difficult job.

The base verb here is uzlaşmak, which means to reach a mutual agreement or to compromise. The -laş part gives it a reciprocal or mutual feeling. By adding the causative suffix -tır, you get uzlaştırmak, which literally means to cause them to reach a mutual agreement. In English, we translate this causative-reciprocal stack simply as to reconcile.

Why does şirket take the accusative case (şirketi) here? Doesn't compromising usually take -le (with)?

If the verb were just the reciprocal uzlaşmak (to compromise), it would indeed take the instrumental case, like in şirketle uzlaşmak (to compromise with the company). However, because we stacked the causative -tır onto it, the companies are no longer the subjects doing the action—they are the direct objects receiving the action of being made to agree. This makes İki rakip şirketi the direct object of uzlaştırmak, requiring the accusative suffix -i.

In the phrase iki rakip şirketi, why isn't it a compound noun like rakip şirketleri?

Here, rakip is acting as an adjective meaning rival or competing, directly modifying şirket (company). If we used the compound noun structure like rakip şirketleri, it would mean something closer to companies of a competitor. As an adjective, it just sits next to the noun without requiring a possessive suffix on şirket.