What are the "ex" links? Why do some e-hentai tags exist, but have no galleries tagged with them?
Some of more questionable for advertisers content is not available on the main e-hentai.org site for most users, but is available on eXhentai. The exact algorithm for obtaining access to it is a mystery, however any of these steps may help:
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have an actively used account that is at least a week old
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upload valuable content to the site (avoid uploading content from the DNP list and DMCA-happy creators, and the questionable content (shota/loli/beastiality/etc.) — the latter is likely already uploaded, and you have no reliable way to verify it yet)
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make a one-time donation equivalent to $20 or more (Bronze star)
Once you think you are eligible, clear all cookies for all e-hentai related domains (*.e-hentai.org, *.exhentai.org, hentaiverse.org, *.hath.network), and then log in to the e-h. Only after a successful login you may attempt to open eX again.
In case of failure try again in a few days. The cookie cleaning procedure will need to be repeated as well.
Avoid browser extensions claiming the ability to grant you instant eX access and importing other people's cookies. These solutions are extremely unreliable and may even cause your account suspension.
Note: unconfirmed reports state that new accounts registered from some countries may never be eligible to access eX. In this case you'll have to use a VPN or other kind of proxy to register one.