Many mobile carriers block text messages that include links shortened with publicly-accessible link shorteners. Links with the bit.ly domain may be marked as spam. To avoid this, use a custom domain instead. A short domain that includes your brand is much less likely to be blocked by mobile carriers.
Visit the Bitly Help Center guide to custom domains for help getting started with custom domains.
Different carriers have different policies about what links and messages they block. For additional information on a specific carrier's policies, contact them or visit their help center.