Giving several vaccines at one appointment is routine and well supported by evidence. The immune system handles many antigens every day, and combining vaccines does not reduce their effectiveness or cause more serious side effects. Different injections are simply given in different sites, such as separate arms.
Co-administration is common for adults during respiratory season, for example flu with COVID-19, and for children following their schedule. There are a few exceptions and spacing rules for certain live vaccines or specific products, which is why a clinician checks your history first.
Getting vaccines together can mean slightly more local soreness or a bigger chance of feeling tired or feverish for a day, but this is usually mild. If you are pregnant, immunocompromised, or unsure about timing, ask a doctor, nurse, or pharmacist to confirm which combinations are right for you.