Just the opposite for me. Every time I tried to use my CC online to make a purchase over $1000, it gets flagged for fraud. Never once with my debit card, and I have purchased a $5000 OLED tv with it online, all my iPhones with it online, and no problems. I tried to purchase my iMac earlier this year online from Apple using my CC and it was denied for possible fraud. Used my debit card, and went through without issue. My credit rating is 850, so it's not bad credit.
Your bank might have a different threshold, however most banks use a lower threshold on debit cards as Teton to reverse a debit is more of a pain than credit.