Federal and state refunds come from completely separate entities, There is no rule as to which one will come first or how much time there will be between their arrivals.
Make sure your state return was accepted: https://turbotax.intuit.com/tax-tools/efile-status-lookup/
To track your state refund:
You asked about state but then said IRS. If you mailed your returns did you mail state in a separate envelope to the state? Some people have been mailing both federal and state in the same envelope just to the IRS.