Pronouncing th after r in Standard American English: /ɹð/

I natively speak Flemish (Dutch). We trill the R.

I just had a 7-lesson course (over video chat with an American lady) to improve my accent towards Standard American English. According to the test I improved greatly, but there is just one particular sound I still have great trouble with.

If an r gets anywhere near a th, I get a tongue tap (as in /R/). I can now usually manage to get thr right, but r th still often fails, at least if the th is a /ð/. Earth and north are no trouble, but for the is harder, and one I can NEVER even get right (at speed) is bother them. I've heard examples of people saying those words right after each other while still clearly pronouncing that /ɹ/ and /ð/, and I want to know how!

My trouble with bother them is that after the first th, my tongue is moving backwards to produce the r, and then it has to travel too quickly back to the front for the second th, tapping my palate along the way. It seems physically impossible to avoid that.

It's probably mostly related to how I pronounce /ɹ/. My previous habit was to twist my tongue (placing it sideways, touching both top and bottom of the mouth with either sides). While this actually sounded fine, I figured learning the 'right way' might improve my chances of pronouncing composite consonants more correctly. So I'm now trying to always pronounce it by having the sides or my tongue press against my upper molars instead, pointing the tip upwards but not touching. That "the way", right?

So in short:

I would value any very specific tips/explanations on pronouncing bother them correctly, i.e. with /ɹð/ but without tapping the palate.

A few points come to mind (as a linguist, but native speaker of a nonrhotic dialect):

1. Pronounce ɹ correctly, in the way that you’ve recently discovered: symmetric on left and right sides; no sideways twisting—rotation isn’t rhotation ;-)
2. When I pronounce ð and θ, they are not always apicodental (tip of the tongue against the teeth), but often approach laminodentals (blade of tongue against teeth). However, after n and ɹ, I have to make them strict apicodentals. So, check you're doing that.
3. Practise saying bother no one. To produce ɹn, your tongue makes a similar transition as in ɹð, ɹθ, but it’s shorter. So, it might be easier to wrap your tongue around (pardon the pun). Likwise: bother Danes/Thais.
4. Then practise they’re there. Same transition as the one you’re interested in, but the longer vowel may ease the transition. You may also find that you can make the apicodentals barely more than a tap.

After that, hopefully bother them will be less bothersome.

Really interesting. I believe a Flemish /r/ is more like an Italian or Spanish one, so [ɾ] or even [r] itself, depending. That means for you it touches the alveolar or dento-alveolar ridge.

And that’s the whole problem. You can’t do that in English. Our /r/ is strictly a non-touching sound.

The English phonemic /r/ is actually an approximant, either an alveolar [ɹ] or especially in America a retroflex one like [ɻ], so you should not be tapping at all. Your tongue does not belong anywhere near touching anything. That seems to be your flaw.

And when it starts a word, like ring or round, our /r/ is often rounded, too, so [ɻʷ].

In any case, being an approximant, it’s a far “squishier” sound than you are used to. There should be no contact, just a curled up tongue in the back of your mouth. In several ways, it’s completely opposite to your own native sound for that letter.