Cloud-computing web systems and services revolutionized the web. Nowadays, they are the most important part of the Internet. Cloud-computing systems provide the opportunity for businesses to undergo digital transformation in order to improve efficiency and reduce costs. The sudden shutdown of schools and offices during the pandemic of Covid 19 significantly increased the demand for cloud solutions. Load balancing and sharing mechanisms are implemented in order to reduce the costs and increase the quality of web service. The usage of those methods with adaptive intelligent algorithms can deliver the highest and a predictable quality of service. In this article, a new HTTP request-distribution method in a two-layer architecture of a cluster-based web system is presented. This method allows for the provision of efficient processing and predictable quality by servicing requests in adopted time constraints. The proposed decision algorithms utilize fuzzy-neural models allowing service times to be estimated. This article provides a description of this new solution. It also contains the results of experiments in which the proposed method is compared with other intelligent approaches such as Fuzzy-Neural Request Distribution, and distribution methods often used in production systems.