I have a website with a good amount of traffic but I must shut down this website.
I have another website that is not very well ranked on Google.
Is there something I could do to help my second website with SEO such as a redirect or DNS change?
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I am not sure why you need to shutdown you site, but if that includes giving up your domain name or host, then no there is nothing you can do.
However, if you can retain your domain name and still host it somewhere, then a blanket 301 redirect will preserve some of your back link status and preserve the domain site metrics. If you can redirect some or as many you can of the older pages to a page of a similar/complementary topic on the new site, then that is best. This preserves nearly all of the value of the inbound (back link) to your original site and will pass it to the new site. Otherwise, search engines will compare the from and to pages of any link and redirect and if the topics do not match, there will be some loss of value and may be discounted almost entirely in time.
Redirection is the way to send both users and search engines to a different URL but from the directly requested URL. Redirection is a very common practice and it is basically about the direction of pages from one page to another. By doing such a practice one can maintain the SEO value of the site.Hence the idea of redirection is quite good as it will move all the traffic of previous site to the new one which will create a good SEO value of your new site.