There are some anime original series but we'll focus on those that were based on manga.
Manga is released before an anime because an anime is an adaptation of the manga. Look here, if we do as you say and stop publishing manga the moment an anime series is started, then how are the animation team supposed to continue the story? Their script and everything that happens in the anime is based on what happens in the manga. Writing a script is not the same as making a manga. A mangaka is not the same as a scriptwriter.
Second, the source material do not adjust for its anime. Rather, the anime adjusts for its source material. There are anime that did not get enough episodes or whose next seasons were delayed because it was catching up to the manga. A good example is Attack on Titan. Here, the mangaka was not forced to speed up release to continue the anime. Rather, they waited for Isayama's story to develop. In Tokyo Ghoul, the second season had a different ending but this did not affect nor force the mangaka to change or adjust his manga's ending or sequel.
Third, I don't get the point of the mangaka being forced to stop publishing their story once it gets an anime adaptation. A manga or light novel being adapted to an anime is usually an indicator that the series is popular. All the more reason for them to continue the manga and maintain the readers' interests as this would contribute to its chances getting another season if it remains popular. Also, I highly doubt that publishers would allow a popular series to stop publishing just because it's getting an anime adaptation. That would drive away consumers which is kind of counter-intuitive.