Usually you can — culinary amounts of turmeric and short-term ibuprofen aren't flagged as a dangerous pair by health authorities, but ibuprofen carries its own stomach and bleeding risks, so use the lowest effective ibuprofen dose, take it with food, and ask your pharmacist before adding turmeric supplements.
No health authority lists a specific, named turmeric–ibuprofen interaction. NCCIH says conventionally formulated oral turmeric/curcumin (not modified to enhance bioavailability) is likely safe in the recommended amounts for up to 2 or 3 months, but warns it can cause stomach upset (nausea, acid reflux, diarrhea) and that "if you take any type of medicine, talk with your health care provider before using any herbal product; some herbs and medicines interact in harmful ways." The real concern is additive: ibuprofen is an NSAID that itself irritates the stomach lining and can cause ulcers and GI bleeding, and the NHS warns not to take ibuprofen alongside other things that raise bleeding risk such as anticoagulants like warfarin, aspirin or other NSAIDs — so combining it with a supplement is something to clear with a pharmacist or doctor rather than assume is fine.
Use ibuprofen at the smallest dose for the shortest time and take it with or after food, and do not exceed the dose on the label unless a doctor directs otherwise. Stop and get urgent help (in the US call your doctor or 911; the NHS advises calling 111) for signs of a stomach bleed: stools that are black, sticky and tar-like, vomiting blood or material that looks like coffee grounds, or severe stomach pain. Be more cautious — and talk to a pharmacist or doctor first — if you have had a stomach ulcer or GI bleed, have a heart, liver or kidney condition, or take a blood thinner (e.g. warfarin), aspirin or another NSAID, since these raise the risk of bleeding; older adults are also generally at higher NSAID risk. Choose food-level turmeric or conventional (not high-bioavailability) supplements, since NCCIH notes liver damage has been reported in some people who took enhanced-bioavailability curcumin products — stop and call your provider if you notice fatigue, nausea, poor appetite, dark urine, or yellowing of the skin or eyes. If you find yourself needing ibuprofen regularly, see a clinician rather than relying on long-term self-treatment.
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.