Whenever I pronounce it, it seems as if my tongue says "What is dis ting" (Albeit, there's a slight "h" in the thing). Is it normal for this to happen?
It is not abnormal in casual speech, or if you speak a dialect that does not distinguish the phonemes d-ð and t-θ.
If your pronunciation of "d" for "ð" or "t" for "θ" draws unwanted attention to you, you should practice saying the sounds as in standard, formal speech.
Here is an article that might help.The ABA Journal
Even if there is no known problem with how you pronounce these sounds, it would be good to be able to distinguish the d-ð and t-θ pairs when you need to do so. That would mean practicing the sounds. Pronouncing the sounds correctly in formal speech can be a benefit to anybody.
A general summary of the issues is here:Wikipedia
To many of us the fact the fricative forms of "d" and "t" still exist in English is a mystery. The Germanic Languages on Continental Europe have generally lost these sounds. That they still exist in Icelandic is not so much a mystery, as that nation has been so isolated from outside spoken language influences, at least until recently. But English has had considerable influence from the Continent the last thousand years, the last four hundred with those sounds gone from Continental Germanic.
Not "normal" sounding in English. If you pronounce "th" like a "d" then you are just having an accent to a native English speaker. We can still understand you, but it isn't proper English pronunciation. "Th" always needs the tongue between the teeth to make the correct sound. I don't know where you're from, but if you are a Spanish speaker, it would be like pronouncing "ñ" the same as "n", if that helps.