Not on your own — ashwagandha can nudge your thyroid hormone levels and may interact with thyroid medication, so the NIH/NCCIH advises against it for people with thyroid disorders; only combine them with your doctor's sign-off and monitoring.
Ashwagandha appears to stimulate the thyroid: in human studies it has raised T3 and T4 and lowered TSH (NIH Office of Dietary Supplements) — the same direction your thyroid medication works. Because of this, NCCIH (part of NIH) explicitly states ashwagandha "might interact with... thyroid hormone medications" and is "not recommended for... those who have autoimmune or thyroid disorders." Stacking it on top of levothyroxine or other thyroid drugs could push your hormone levels too high (toward hyperthyroidism) — NIH describes case reports of thyrotoxicosis in people taking ashwagandha that resolved when they stopped. The effect varies by person and dose, which is why this is a decision to make with your clinician rather than on your own.
Talk to your doctor or pharmacist before adding ashwagandha if you take levothyroxine, liothyronine, or desiccated thyroid — they may want to check your thyroid blood work (TSH, T3, T4) and adjust your dose. Watch for signs your thyroid level has gone too high: rapid or pounding heartbeat, anxiety or jitteriness, tremor, trouble sleeping, unexplained weight loss, or feeling overheated — and call your provider if these appear. NCCIH says to avoid ashwagandha if you have an autoimmune or thyroid disorder; NIH also advises caution if you are pregnant or breastfeeding or have liver problems, since there are rare reports of ashwagandha-linked liver injury. This is separate from absorption-timing rules (calcium, iron, kelp) on the NHS levothyroxine page; tell your pharmacist about any herbal supplement you're considering. Do not start, stop, or change either one without medical advice.
This is general reference, not medical advice, and not a guarantee of safety. Interactions depend on your doses, health conditions, and other medicines. Always confirm with your pharmacist or doctor before combining products, and follow the dosing on each label.