Can she renew the ISA with the £20,000 plus interest to a new fixed term ISA
That will depend on the providers rules. It is likely that it can be transferred into another Fixed Term ISA and not count as new money going in as it is already Inside the ISA wrapper.
Then add the £5,000 plus interest to it?
Yes, that would be new money going in to the ISA wrapper this tax year. A fixed term ISA likely has rules about when payments can go in, such as during first 30 days of opening the account.
Will she get taxed on anything over £20,000?
Generally I will say no. If the money is all inside ISA wrapper, then what is inside is tax free. Interest on the £5k that was outside the ISA would be subject to tax in the tax year that the interest was paid. So it might have spread the interest payments over last tax year and this tax year. Likely to be no tax on it, as there is a Personal Savings Allowance.
The bank seems to be saying you can only ever have £20,000 in an ISA that is free from tax and the rest you pay tax on.
That sounds wrong. The ISA allowance applies to new money going in to ISA (and LISA) in the tax year. Money already in an ISA which is from a previous tax year, is not new money going in.
Think about:
What is the money for? If some will be used towards a first home purchase, then consider using a Lifetime ISA (max £4k can be paid in during a tax year).
When will they need access to the money? Is locking it away sensible? If it will not be accessed for over 5 years, then could it be invested?