Why is 两十万 (liǎng shí wàn) okay when it contains 两十 (liǎng shí) which is incorrect?
「兩」didn't originally mean two. The character was first used for what is now written「輛」, the unit of measure for vehicles, being differentiated from「丙」and composed of two「丙」side-by-side:



The mark「一」was a decorative stroke added on top later to produce「兩」(note, a similar process occurred with the character「雨」).

The etymology of liǎng is ultimately not known, but very early on,「兩」was used as a phonetic loan for several meanings. The relevant ones to the modern language are:


Comparing and contrasting, or describing, two objects/people/situations.


《詩・齊風・還》： “並驅從兩肩兮， 揖我謂我儇兮。” 
《莊子・人間世》： “吾未至乎事之情， 而既有陰陽之患矣；事若不成， 必有人道之患。 是兩也， 為人臣者不足以任之， 子其有以語我來。” 

Unit of measurement of weight or objects.


《詩・齊風・南山》： “葛屨五兩， 冠緌雙止。”
《禮記・雜記下》： “納幣一束， 束五兩， 兩五尋。” 



From traditional usage,「兩」(1) first of all refers to one or several units of something, then by extension (2) refers to a comparison or description of several/some/a small number of or two units/things. This is not the same usage as the abstract number「二」, and is why you never say abstract numbers with「兩」(e.g. when talking about telephone numbers, decimal numbers, etc.), and never use「兩」when counting objects which use「兩」as a measure word (e.g. for taels).

It is this differential usage which also precludes「兩十」from being a clear expression; 「十」is homophonous with too many other words, unlike「百」,「千」,「萬」.

To answer the question directly,


「兩十」also turns up Baidu hits; it is just as valid or invalid as「兩十萬」
Some of the Baidu hits are actually using「兩」as「輛」
The other ones may be regional colloquialisms; from a discussion here:



  我們老家還眞有，兩個，一十兩，兩十，兩百零兩，都是50歲以上的人，現在已經不怎麽能聽到這種數字了，方言被普通話漸漸消滅了。。。




Finally, note that


In formal currency numerals, there is only the alternative character「貳」, and none corresponding for「兩」. This is related to「兩」being invalid for use as a number in traditional currency systems.
It is exceedingly rare to use「兩」for the meaning two in the Sinoxenic languages (Japanese, Korean, Vietnamese), indicating that using it this way freely is a rather recent colloquial Chinese invention