All parts of the Horseradish contain glucosinolates (although the greatest concentration is in the root), which can cause stomach problems in mammals.
Although safe for tortoises, this plant is also a member of the Brassica family and contains goitrogens, so it should be fed in limited quantities only. Never feed the root of the Horseradish.
Please be aware that two or more goitrogenic plants should not be fed in conjunction with each other, and the sum total of goitrogenic foods should not exceed the guideline for a single plant of 'in moderation' (amber-green) or 'sparingly' (amber-red). This will ensure that goitrogenic consumption stays within safe limits.